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Z>U0P_}I?:w݋4^\MZ{@i%~S?UuV@-}ƞL6XjwwE_(s<]W[7[W4IڶƗ[+HGݑWwS5r _l~.C t]̕DO#ÿe =sNa߻M+t%S59eo5kslJ|w$#O{=<~|^Cf߹(yp S};鷥Mɡ-o}ɽf Nuzlݚ"jE\x ~f,ո?ޞf`L_RHKផ@m$NtIlPH|wP=,2w= nznFu+[r@-t elB=;FT?Hw2[nLĦ}+]E*./*jw[&x5jo&fwI _{<@ܜ8oͦ_>jXXgkmck0)f?!D:@;mS}pJczhKClEc41Ȑm )'KRf3 J'+=J omqga=u\8(VN2{|f?T%\>[J4v}(=T-Fʱ dA+W(?1psg骺/^.$3grk[9eq"{z=̛F0n>ݫv}V94S&j> 4H!Lܢls=FOS&9K[-&,k aGu(V[H "Da+iRp k+g8 X 6 [i2p4@ R_8I]cT;E 'y +鼕,.>Uϲ`XS+4,"ӚFx(&.T{'皆hIPwx(ҡ:Z,wlN##,-lQsִ!%1)k/cQʰ~s5T1bQf|LwbA FY9v t `GJJhUUr s<̄_$@"cg 5D٩^#QD'V`a8&e jD kMk.4YbK-å^ʉ2ֹr @̢\@C9T.~T { [MSl>mRO1-xՕͬp>QRڀ%f42P^{fu=$l8v,0{Qm~],'Z^a(UuERCNJsTRP\\" \?ʨϘ,NUtI1lG^v6ɾEu^͓5*ya͟2O=W9~ƕHY׃u T)լ Bc`XJZN-c &F z~=iGⓢ͖akv vÃ2%ڴ5 Dʤࡐ  }32-NbJjm ]ȗS].0+m޲瑙`Sās錎qǰu 7`W507:ĭVxZL8qZB`\ssЍw!y\'kZȂ>RJ yrގ ue M()rN&|xe$4s-D͍\#DFʀDX`1xV O-3~4!okͫ -X:<,u5XXLt%޲CDN|!TVrZȦwqAm{~z4%n"6FW\k0.Ľ: ~ip؛2<=}N %vзfXۧݼr;!WCc#n Ns%dG(=um(C푆gnU]giB^,_6$q1=Nf۸<çaaa$Jz8_ }ٔk~_I [UA/7koF߬{NMы-TOuFd,kzpUX+V \Z˭qƧr$&l^6\J(g*Ehzz @?5n}Y;w#ln޺ºM7 J(ձ2[嬸C>+P]9Yg4\6&i̛l/OR|/$ fI nl>cTkUQbGEx|>ra_{8 " 0 "" {MPLTEXT 1 0 51 "simplify(%) assuming phi in RealRange(-Pi/2, Pi/2);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticFG6$-%$sinG6#%$phiG%\"kG-%$IntG6$*$,(\"\"\"F1*$F+\" \"#!\"\"*&F+F3-%$cosG6#%(_theta1GF3F1#F4F3/F9;\"\"!F* "}}} {PARA 0 "> " 0 "" {TEXT -1 70 "where the right-hand-side is actually equal to the trigonometric form " } {TEXT 35 22 "InverseJacobiAM(phi,k)" } {TEXT -1 147 ". The general relationship between these two forms and the restriction on the values of the parameters such that the relation is valid are given by" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "FunctionAdvisor( specialize, InverseJacobiAM, EllipticF);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/-%0InverseJacobiAMG6$%$phiG%\"kG-%*EllipticFG6$-%$sinG6# F(F)55-%$AndG6$2,$%#PiG#!\"\"\"\"#-%#ReGF/2F;,$F7#\"\"\"F:-F36$/F6F ;1\"\"!-%#ImGF/-F36$/F>F;1FFFE "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "FunctionAdvisor( specialize, EllipticF, InverseJacobiAM);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/-%*EllipticFG6$%\"zG%\"kG-%0InverseJacobiAMG6$-%'arcsinG6 #F(F)-%9with~no~restrictions~on~G6$F)F( "}}} } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {PARA 0 "> " 0 "" {TEXT -1 43 "Reflection symmetry and special values for " } {TEXT 35 9 "EllipticK" } {TEXT -1 5 " and " } {TEXT 35 9 "EllipticF" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "FunctionAdvisor( special_values, EllipticK);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7&/-%*EllipticKG6#,$%\"kG!\"\"-F&6#F)/-F&6#\"\"!,$%#PiG#\"\"\"\"\"#/ -F&6#%)infinityGF0/-F&6#^#F9F0 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "FunctionAdvisor( special_values, EllipticF);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7(/-%*EllipticFG6$,$%\"zG!\"\"%\"kG,$-F&6$F)F+F*/-F&6$F),$F+F* F-/-F&6$\"\"!F+F6/-F&6$\"\"\"F+-%*EllipticKG6#F+/-F&6$F)F6-%'arcs inG6#F)/-F&6$F)F:-%(arctanhGFC "}}} {PARA 0 "> " 0 "" {TEXT -1 18 "Branch points for " } {TEXT 35 9 "EllipticF" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "FunctionAdvisor( branch_points, EllipticF);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$-%*EllipticFG6$%\"zG%\"kG-%#inG6$F'7'!\"\"\"\"\",$*$F(F-F-F0^$% )infinityGF2 "}}} {PARA 0 "> " 0 "" {TEXT -1 37 "Branch points and the branch cut for " } {TEXT 35 9 "EllipticK" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "FunctionAdvisor( branch_points, EllipticK);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$-%*EllipticKG6#%\"kG-%#inG6$F'7&^$,$%)infinityG!\"\"F-F/\"\"\" ^$F.F. "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "FunctionAdvisor( branch_cuts, EllipticK);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7%-%*EllipticKG6#%\"kG2F'!\"\"2\"\"\"F' "}}} {PARA 0 "> " 0 "" {TEXT -1 4 "For " } {TEXT 35 5 "Re(k)" } {TEXT -1 20 " in the cut, so for " } {TEXT 35 22 "1 <= Re(k) <= infinity" } {TEXT -1 2 ", " } {TEXT 35 9 "EllipticK" } {TEXT -1 26 " is continuous from below." } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "EllipticK(2 - 1/100000 * I): # Im(k) < 0 and Re(k) in RealRange(1,infinity),\n% = evalf(%);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticKG6#^$\"\"##!\"\"\"'++5^$$\"+*G$yG%)!#5$!+KHDy5!\"* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "EllipticK(2): # k in the cut, Im( K(k) ) is continuous from below\n% = evalf(%);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticKG6#\"\"#^$$\"+u " 0 "" {MPLTEXT 1 0 100 "EllipticK(2 + 1/100000 * I): # Im(k) > 0, there is a jump in the value of Im( K(k) )\n% = evalf(%);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticKG6#^$\"\"##\"\"\"\"'++5^$$\"+*G$yG%)!#5$\"+KHDy5!\"* "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "RealRange" 2 "RealRange" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticE" 2 "EllipticE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCE" 2 "EllipticCE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticPi" 2 "EllipticPi" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCPi" 2 "EllipticCPi" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "FunctionAdvisor" 2 "FunctionAdvisor" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobiAM" 2 "JacobiAM" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "InverseJacobiAM" 2 "InverseJacobiAM" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "WeierstrassP" 2 "WeierstrassP" "" }{TEXT -1 2 ". " } } }} " } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticKG6#\"\"#^$$\"+u " 0 "" {MPLTEXT 1 0 100 "EllipticK(2 + 1/100000 * I): # Im(k) > 0, there is a jump in the value of Im( K(k) )\n% = evalf(%);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticKG6#^$\"\"##\"\"\"\"'++5^$$\"+*G$yG%)!#5$\"+KHDy5!\"* "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "RealRange" 2 "RealRange" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticE" 2 "EllipticE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCE" 2 "EllipticCE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticPi" 2 "EllipticPi" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCPi" 2 "EllipticCPi" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "FunctionAdvisor" 2 "FunctionAdvisor" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobiAM" 2 -1 49 "procedure or expression representing the function" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "x - " } {TEXT -1 51 "variable name appearing in f, if f is an expression" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "a, b - " } {TEXT -1 57 "numerical values specifying the interval of approximation" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "eps - " } {TEXT -1 24 "(optional) numeric value" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 57 "This function computes the Chebyshev series expansion of " } {TEXT 35 1 "f" } {TEXT -1 31 ", with respect to the variable " } {TEXT 35 1 "x" } {TEXT -1 17 " on the interval " } {TEXT 35 4 "a..b" } {TEXT -1 20 ", valid to accuracy " } {TEXT 35 3 "eps" } {TEXT -1 2 ". " } } {PARA 15 "> " 0 "" {TEXT -1 40 "If the second argument is simply a name " } {TEXT 35 1 "x" } {TEXT -1 19 " then the equation " } {TEXT 35 9 "x = -1..1" } {TEXT -1 13 " is implied. " } } {PARA 15 "> " 0 "" {TEXT -1 25g>ARA 15 "> " 0 "" {TEXT -1 117 "However, users who get used to ending all statements with a semicolon are accommodated since, if a semicolon follows " } {TEXT 35 3 "i^3" } {TEXT -1 93 " here, then in the view of the Maple parser there is an empty statement prior to the keyword " } {TEXT 35 6 "end do" } {TEXT -1 2 ". " } } }} {TEXT 35 4 "null" } {TEXT -1 2 "; " } } {PARA 15 "> " 0 "" {TEXT -1 117 "An empty statement in Maple contains no command at all. For example, the empty statement is the second statement in: " } } {PARA 0 "> " 0 "" {TEXT 23 19 " a:=1; ;quit" } } {PARA 15 "> " 0 "" {TEXT -1 214 "The main purpose of the empty statement is to allow redundant semicolons to be used without causing a syntax error. For example, in strict Maple syntax there would be no semicolon statement separator following the " } {TEXT 35 4 "cube" } {TEXT -1 18 " statement below: " } } {PARA 0 "> " 0 "" {TEXT 23 91 " for i to 5 do\n sqr[i] := i^2;\n cube[i] := i^3\n end do;" } } {Pnctfunctindexellipticellfellipticckelliptickellipticfitcutsinfinitygcutpointbrancharctanhgfcinginfinitygfspecialsymmetrreflectfffeimgfregfpigandgspecializgivenvalidsuchvalurelationshipgeneraltrigonometricactualsidehandrightcosgrealrangassumsimplifarcscosgfdchangpdetoolsinvariablchangeaseformpiinstancthetasingphiginversejacobiamgintconvertphiinversejacobiamabovlineintegranddenominatorrootsplittbutbetweendifferencnotingworthcommandsqrtequalnumericalshownexamplmaplentersquarrepresenttermthesstegunabramowitzeditmathematicalhandbookpopularryzhikgradshteynproductseritabldescribwellholdparametersomeprovidfunctionallrepresusedcanthereforinversejacobisngfinversejacobisnidenticalalsorelatstrictreellipticckggfelliptickgrespectiverestrictnowithalphaintgkgzgellipticfgdefinitfunctionadvisordefinparametamplitudsineexpressalgebraiccomplementarcompletkindfirstintegralincompletconvertfu " 0 "" {TEXT -1 9 "ellipsoid" } {TEXT 30 31 " - surface area of an ellipsoid" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 15 " ellipsoid(" } {TEXT 35 1 "a" } {TEXT -1 2 ", " } {TEXT 35 1 "b" } {TEXT -1 2 ", " } {TEXT 35 1 "c" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 8 "a,b,c - " } {TEXT -1 47 "the three principal semi-axes of the ellipsoid " } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 14 "The procedure " } {TEXT 35 9 "ellipsoid" } {TEXT -1 62 " returns the surface area of an ellipsoid whose semi-axes are " } {TEXT 35 1 "a" } {TEXT -1 2 ", " } {TEXT 35 1 "b" } {TEXT -1 6 ", and " } {TEXT 35 1 "c" } {TEXT -1 7 " where " } {TEXT 35 6 "a >= b" } {TEXT -1 2 ", " } {TEXT 35 6 "b >= c" } {TEXT -1 2 ", " } {TEXT 35 6 "c >= 0" } {TEXT -1 89 ". For a general ellipsoid the result is expressed as an unevaluated integral in terms of " } {TEXT 35 3 "Int" } {TEXT -1 41 " , which may be numerically evaluated by " } {TEXT 35 5 "evalf" } {TEXT -1 2 ". " } } {PARA 15 "> " 0 "" {TEXT -1 162 "When two or more of the arguments are the same, then a solvable integral can be created. Otherwise, elliptical integrals are needed to evaluate the general case. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(ellipsoid(5,3,2));" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+n " 0 "" {TEXT -1 1 " " } } }} A 15 "> " 0 "" {TEXT -1 14 "The procedure " } {TEXT 35 9 "ellipsoid" } {TEXT -1 62 " returns the surface area of an ellipsoid whose semi-axes are " } {TEXT 35 1 "a" } {TEXT -1 2 ", " } {TEXT 35 1 "b" } {TEXT -1 6 ", and " } {TEXT 35 1 "c" } {TEXT -1 7 " where " } {TEXT 35 6 "a >= b" } {TEXT -1 2 ", " } {on " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "a - " } {TEXT -1 11 "expression " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "eqns - " } {TEXT -1 24 "list or set of equations" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "n - " } {TEXT -1 17 "positive integer " } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 27 "The most common use of the " } {TEXT 35 4 "eval" } {TEXT -1 39 " function is to evaluate an expression " } {TEXT 35 1 "e" } {TEXT -1 18 " at a given point " } {TEXT 35 3 "x=a" } {TEXT -1 15 ". For example, " } {TEXT 35 20 "eval(x^2+3*x+2,x=1);" } {TEXT -1 26 " evaluates the polynomial " } {TEXT 35 9 "x^2+3*x+2" } {TEXT -1 14 " at the point " } {TEXT 35 3 "x=1" } {TEXT -1 14 " obtaining 6. " } } {PARA 15 "> " 0 "" {TEXT -1 38 "In the second calling sequence, where " } {TEXT 35 4 "eqns" } {TEXT -1 47 " is a set or list of equations, the expression " } {TEXT 35 1 "e" } {TEXT -1 71 " is evaluated at all the given pointion points" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 4 "Eval" } {TEXT -1 70 " function is a place holder for evaluation at a point. The expression " } {TEXT 35 1 "a" } {TEXT -1 17 " is evaluated at " } {TEXT 35 3 "x=n" } {TEXT -1 2 " (" } {TEXT 35 5 "x1=n1" } {TEXT -1 2 ", " } {TEXT 35 5 "x2=n2" } {TEXT -1 34 ", ... for the multivariate case). " } } {PARA 15 "> " 0 "" {TEXT -1 93 "To perform an evaluation of an expression or a polynomial over the rational numbers, use the " } {HYPERLNK 17 "eval" 2 "eval" "" }{TEXT -1 10 " command. " } } {PARA 15 "> " 0 "" {TEXT -1 9 "The call " } {TEXT 35 18 "Eval(a, x=n) mod p" } {TEXT -1 26 " evaluates the polynomial " } {TEXT 35 1 "a" } {TEXT -1 4 " at " } {TEXT 35 3 "x=n" } {TEXT -1 8 " modulo " } {TEXT 35 1 "p" } {TEXT -1 18 " . The polynomial " } {TEXT 35 1 "a" } {TEXT -1 56 " must be a multivariate polynomial over a finite field. " } } {PARA 15 "> " 0 "" {TEXT -1 9 "The call " } {TEXT 35 20 "mre root of a quadratic polynomial in " } {TEXT 35 3 "sin" } {TEXT -1 5 " and " } {TEXT 35 3 "cos" } {TEXT -1 29 ") and hyperbolic trig forms. " } } {PARA 15 "> " 0 "" {TEXT -1 121 "Elliptic integrals are reduced to their Legendre normal form in terms of elementary functions and the Elliptic functions " } {TEXT 35 9 "EllipticF" } {TEXT -1 2 ", " } {TEXT 35 9 "EllipticE" } {TEXT -1 6 ", and " } {TEXT 35 10 "EllipticPi" } {TEXT -1 31 " (or their complete versions). " } } {PARA 15 "> " 0 "" {TEXT -1 12 "REFERENCES: " } } {PARA 0 "> " 0 "" {TEXT 23 177 " G. Labahn and M. Mutrie, Reduction of Elliptic Integrals to Legendre\n Normal Form, University of Waterloo Tech Report, Department of\n Computer Science 97-21" } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {PARA 0 "> " 0 "" {TEXT -1 18 "Elementary answer " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int( sqrt(1+x^4)/(1-x^4),x=0..1/2);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,**&\"\"##\"\"\"F%-%#lnG6#,&!\"%F'*&\"# " 0 "" {TEXT -1 20 "Symbolic parameters " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "assume(0 " 1 "" {XPPMATH 20 "6#,&*&%#k|irG!\"#-%*EllipticFG6$F%F%\"\"\"F**&F%F&-%*EllipticEGF )F*!\"\" "}}} {PARA 0 "> " 0 "" {TEXT -1 23 "Answer as sum of roots " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ans := int(1/(x^4 + 2)/sqrt( 4 - 5*x^2 + x^4),x=0..1/4);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%$ansG,$-%$SumG6$-%+EllipticPiG6%#\"\"\"\"\"%*$%'_alphaG!\"##F- \"\"#/F0-%'RootOfG6#,&*$%#_ZGF.F-F3F-#F-\"#; "}}} {PARA 0 "> " 0 "" {TEXT -1 32 "Can evaluate to floating point: " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(ans);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#^$$\"++r?Jj!#6$\"\"!F( "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ans,20);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#^$$\"5QF*R<=527L'!#@$\"\"!F( "}}} {PARA 0 "> " 0 "" {TEXT -1 10 "Trig form " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "int( sqrt( 1 + 2*sin(x) ), x=0..Pi/2) ;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,*-%*EllipticKG6#,$*$\"\"$#\"\"\"\"\"#F*!\"\"-%*EllipticFG6$,$*&F,F *F)F*#F+F)F'F+-%*EllipticEGF&\"\"%-%+EllipticPiG6%F1#F)F6F'F- "}}} {PARA 0 "> " 0 "" {TEXT -1 21 "Indefinite trig form " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(1/sqrt(1 + 2*cos(x) ), x );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,$*0,$*&,&!\"\"\"\"\"*$-%$cosG6#,$%\"xG#F)\"\"#F1\"\"%F),&F(F)F*F)F) F(F0,&F)F)F*F(F0,&F)F)F*!\"%F0,(F(F)*$F+F2F6F*\"\"&#F(F1-%*Elli pticFG6$F+F1F)-%$sinGF-F(F'F:!\"# "}}} {PARA 0 "> " 0 "" {TEXT -1 14 "Check answer: " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "simplify( combine( diff(%,x) - 1/sqrt( 1 + 2*cos(x) ), trig ));" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"\"! "}}} }} 20 "6#^$$\"++r?Jj!#6$\"\"!F( "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ans,20);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#^$$\"5QF*R<=527L'!#@$\"\"!F( "}}} {PARA 04dgfcgeqgeqygfsqrtygxgznammapcoordinatwithdescriptdetailverticalnamfailhorizontalnamcentgeometricreturnfunctuserelatinformataccesaxespromptmaplrespectiveusedwillthesassignenvverticalnamenvhorizontalnamvariablenvironmargumabovexplainformnecessaridoesnoteinputfollowcancallbetweengreatpositgivenfixedwhosplanallsetverticalhorizontalrepresentoptionalequatpolynomialrepresentatalgebraicendanysumminoraxismajorlengthlistdenotlessconstantlinepointdistinctfivenameeqnepdisdistanclmiminoraxilmamajoraxifoifociecceccentricitfoufocudirdirectrixobjectdefinitdefinellipsgeometrpigfxglngexpirgfpigygggfgunprotectlambertwginvfuncglambertwexpglnarcssindoesdirectionalunimaplotheranyjustdeletaddedmayentrnegatnonnumbernaturalnnxxintegerarbitrarstandzxxdomaincorrectassumvariabladditionalnnusesnormalcasevalumultiplreturnbutsimilarcorrespondwithformexpressoccurrencimplicitexplicitreplacoperatorsimplifsolvusednametablfunctinversinvfuncipulatsinglintocombincomplexpartimaginarcopycreatcodeexternaldatarealonlycontainrtabltestrtablezipreimrtablecopyrealpartrtablecopyimpartomvbrtablindexrtableisrealvbopenmaplpolynomg# MO62DY2acobiTheta1)[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%-JacobiTheta1G6$%\"zG%\"qG-%$SumG6$,$*()F(*$,&%$_k1G\"\"\"#F 2\"\"#F2F4F2-%$sinG6#,&*&F'F2F1F2F4F'F2F2)!\"\"F1F2F4/F1;\"\"!%)in finityG "}}} {PARA 0 "> " 0 "" {TEXT 35 15 "EllipticModulus" } {TEXT -1 28 " computes the corresponding " } {TEXT 22 7 "Modulus" } {TEXT -1 1 " " } {TEXT 35 1 "k" } {TEXT -1 2 ", " } {TEXT 35 9 "Re(k) > 0" } {TEXT -1 36 " entering the definition of related " } {HYPERLNK 17 "elliptic integrals" 2 "EllipticF" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "JacobiPQ elliptic functions" 2 "JacobiAM" "" }{TEXT -1 1 "." } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "FunctionAdvisor(definition, EllipticF)[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticFG6$%\"zG%\"kG-%$IntG6$*&,&\"\"\"F.*$%(_alpha1G\"\"#! \"\"#F2F1,&F.F.*&F(F1F0F1F2F3/F0;\"\"!F' "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "FunctionAdvisor(definition, JacobiSN)[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%)JacobiSNG6$%\"zG%\"kG-%$sinG6#-%)JacobiAMGF& "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "FunctionAdvisor(definition, JacobiAM);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/%\"zG-%)JacobiAMG6$-%$IntG6$*$,&\"\"\"F.*&%\"kG\"\"#-%$sinG6#% (_theta1GF1!\"\"#F6F1/F5;\"\"!F%F0'F%-%*RealRangeG6$#!\"$F1#\"\"$F1 "}}} {PARA 0 "> " 0 "" {TEXT -1 33 "Alternatively, given the Modulus " } {TEXT 35 1 "k" } {TEXT -1 2 ", " } {TEXT 35 9 "Re(k) > 0" } {TEXT -1 10 " entering " } {HYPERLNK 17 "Elliptic integrals" 2 "EllipticF" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "JacobiPQ" 2 "JacobiSN" "" }{TEXT -1 61 " functions, it is possible to compute the corresponding Nome " } {TEXT 35 1 "q" } {TEXT -1 2 ", " } {TEXT 35 10 "abs(q) < 1" } {TEXT -1 8 ", using " } {HYPERLNK 17 "EllipticNome" 2 "EllipticNome" "" }{TEXT -1 35 ", which is the inverse function of " } {TEXT 35 15 "EllipticModulus" } {TEXT -1 1 "." } } {PARA 0 "> " 0 "" {TEXT 35 15 "EllipticModulus" } {TEXT -1 24 " is defined in terms of " } {HYPERLNK 17 "JacobiTheta" 2 "JacobiTheta" "" }{TEXT -1 14 " functions by:" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "FunctionAdvisor( definition, EllipticModulus );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/-%0EllipticModulusG6#%\"qG*&-%-JacobiTheta2G6$\"\"!F(\"\"#-% -JacobiTheta3GF,!\"#-%$AndG6#2-%$absGF'\"\"\" "}}} {PARA 0 "> " 0 "" {TEXT -1 4 "The " } {HYPERLNK 17 "JacobiPQ" 2 "JacobiSN" "" }{TEXT -1 40 " functions can be expressed in terms of " } {HYPERLNK 17 "JacobiTheta" 2 "JacobiTheta" "" }{TEXT -1 17 " functions using " } {TEXT 35 12 "EllipticNome" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "JacobiSN(z,k) = (1/(k^2))^(1/4) * JacobiTheta1(1/2*Pi*z/EllipticK(k),EllipticNome(k)) / JacobiTheta4(1/2*Pi*z/EllipticK(k),EllipticNome(k));" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%)JacobiSNG6$%\"zG%\"kG*(*$F(!\"##\"\"\"\"\"%-%-JacobiTheta1G6$, $*(%#PiGF-F'F--%*EllipticKG6#F(!\"\"#F-\"\"#-%-EllipticNomeGF7F- -%-JacobiTheta4GF1F8 "}}} {PARA 0 "> " 0 "" {TEXT -1 34 "Alternative popular notations for " } {HYPERLNK 17 "elliptic integrals" 2 "EllipticF" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "JacobiPQ" 2 "JacobiAM" "" }{TEXT -1 31 " functions involve a parameter " } {TEXT 35 1 "m" } {TEXT -1 20 " or a modular angle " } {TEXT 35 5 "alpha" } {TEXT -1 25 ", as for instance in the " } {TEXT 39 34 "Handbook of Mathematical Functions" } {TEXT -1 61 " edited by Abramowitz and Stegun (A&S). These are related to " } {TEXT 35 1 "k" } {TEXT -1 4 " by " } {TEXT 35 7 "m = k^2" } {TEXT -1 5 " and " } {TEXT 35 14 "sin(alpha) = k" } {TEXT -1 28 ". For example, the Elliptic " } {TEXT 35 4 "K(m)" } {TEXT -1 57 " function shown in A&S is numerically equal to the Maple " } {TEXT 35 20 "EllipticK( sqrt(m) )" } {TEXT -1 9 " command." } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "FunctionAdvisor(definition, EllipticModulus(q) )[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%0EllipticModulusG6#%\"qG*&-%-JacobiTheta2G6$\"\"!F'\"\"#-%-J acobiTheta3GF+!\"# "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf( eval(%, q=1/2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"+6w%*****!#5$\"+ " 0 "" {MPLTEXT 1 0 65 "EllipticModulus( EllipticNome(k) ) = k; # valid for Re(k) > 0" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%0EllipticModulusG6#-%-EllipticNomeG6#%\"kGF* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf( eval(%, k=2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"\"#\"\"!F$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "EllipticNome( EllipticModulus(q) ) = q; # valid for abs(q) < 1" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%-EllipticNomeG6#-%0EllipticModulusG6#%\"qGF* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf( eval(%, q=1/2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"+++++]!#5F$ "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "EllipticF" 2 "EllipticF" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticNome" 2 "EllipticNome" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "FunctionAdvisor" 2 "FunctionAdvisor" "" }{TEXT -1 1 " " } } }} q=1/2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "62{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 34 "Procedure End-of-Parameters Marker" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 161 "Normally the number of arguments in a procedure call need not be the same as the number of specified formal parameters. However, if the end-of-parameters marker " } {TEXT 35 1 "$" } {TEXT -1 149 " is used at the end of the sequence of parameters in the procedure definition, then the passing of any additional arguments will raise an exception. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := proc(a,$) 2*sin(a); end proc;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"aG%#~$G6\"F)F),$-%$sinG6#9$\"\"#F)F)F) "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(Pi);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"\"! "}}}{EXCH stringtool;?v( =   S jX$$ $ t !k"Z$o%%)J,U/ 4D44 5)56;?pBoC?EFG }I4JKL QIS;UU}VXYXZ;[a4b crfg jlnnn #xxy{ 5})~`|/IKQ؈gÑ̔z Ӝp2'(g˸׸". Zg qI;s/Z |=\ition, EllipticF)[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%*EllipticFG6$%\"zG%\"kG-%$IntG6$*&,&\"\"\"F.*$%(_alpha1G\"\"#! \"\"#F2F1,&F.F.*&F(F1F0F1F2F3/F0;\"\"!F' "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "FunctionAdvisor(definition, JacobiSN)[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%)JacobiSNG6$%\"zG%\"kG-%$sinG6#-%)JacobiAMGF& "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "FunctionAdvisor(definition, JacobiAM);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/%\"zG-%)JacobiAMG6$-%$IntG6$*$,&\"\"\"F.*&%\"kG\"\"#-%$sinG6#% (_theta1GF1!\"\"#F6F1/F5;\"\"!F%F0'F%-%*RealRangeG6$#!\"$F1#\"\"$F1 "}}} {PARA 0 "> " 0 "" {TEXT 35 12 "EllipticNome" } {TEXT -1 28 " computes the corresponding " } {TEXT 22 4 "Nome" } {TEXT -1 1 " " } {TEXT 35 1 "q" } {TEXT -1 2 ", " } {TEXT 35 10 "abs(q) < 1" } {TEXT -1 53 ", entering the definition of the related (see below) " } {HYPERLNK 17 "Jacobi Theta" 2 "JacobiTheta1" "" }{TEXT -1 25 " functions, for instance:" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "FunctionAdvisor(definition, JacobiTheta1)[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%-JacobiTheta1G6$%\"zG%\"qG-%$SumG6$,$*()F(*$,&%$_k1G\"\"\"#F 2\"\"#F2F4F2-%$sinG6#,&*&F'F2F1F2F4F'F2F2)!\"\"F1F2F4/F1;\"\"!%)in finityG "}}} {PARA 0 "> " 0 "" {TEXT -1 30 "Alternatively, given the Nome " } {TEXT 35 1 "q" } {TEXT -1 2 ", " } {TEXT 35 10 "abs(q) < 1" } {TEXT -1 54 ", it is possible to compute the corresponding Modulus " } {TEXT 35 1 "k" } {TEXT -1 2 ", " } {TEXT 35 9 "Re(k) > 0" } {TEXT -1 8 ", using " } {HYPERLNK 17 "EllipticModulus" 2 "EllipticModulus" "" }{TEXT -1 35 ", which is the inverse function of " } {TEXT 35 12 "EllipticNome" } {TEXT -1 1 "." } } {PARA 0 "> " 0 "" {TEXT 35 12 "EllipticNome" } {TEXT -1 73 " is defined in terms of the Complete Elliptic integral of the first kind " } {HYPERLNK 17 "EllipticK" 2 "EllipticK" "" }{TEXT -1 4 " by:" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "FunctionAdvisor( definition, EllipticNome );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/-%-EllipticNomeG6#%\"kG-%$expG6#,$*(%#PiG\"\"\"-%+EllipticC KGF'F/-%*EllipticKGF'!\"\"F4-%9with~no~restrictions~on~GF' "}}} {PARA 0 "> " 0 "" {TEXT -1 4 "The " } {HYPERLNK 17 "JacobiPQ" 2 "JacobiSN" "" }{TEXT -1 40 " functions can be expressed in terms of " } {HYPERLNK 17 "JacobiTheta" 2 "JacobiTheta" "" }{TEXT -1 17 " functions using " } {TEXT 35 12 "EllipticNome" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "JacobiSN(z,k) = (1/(k^2))^(1/4) * JacobiTheta1(1/2*Pi*z/EllipticK(k),EllipticNome(k)) / JacobiTheta4(1/2*Pi*z/EllipticK(k),EllipticNome(k));" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%)JacobiSNG6$%\"zG%\"kG*(*$F(!\"##\"\"\"\"\"%-%-JacobiTheta1G6$, $*(%#PiGF-F'F--%*EllipticKG6#F(!\"\"#F-\"\"#-%-EllipticNomeGF7F- -%-JacobiTheta4GF1F8 "}}} {PARA 0 "> " 0 "" {TEXT -1 34 "Alternative popular notations for " } {HYPERLNK 17 "elliptic integrals" 2 "EllipticF" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "JacobiPQ" 2 "JacobiAM" "" }{TEXT -1 31 " functions involve a parameter " } {TEXT 35 1 "m" } {TEXT -1 20 " or a modular angle " } {TEXT 35 5 "alpha" } {TEXT -1 25 ", as for instance in the " } {TEXT 39 34 "Handbook of Mathematical Functions" } {TEXT -1 61 " edited by Abramowitz and Stegun (A&S). These are related to " } {TEXT 35 1 "k" } {TEXT -1 4 " by " } {TEXT 35 7 "m = k^2" } {TEXT -1 5 " and " } {TEXT 35 14 "sin(alpha) = k" } {TEXT -1 28 ". For example, the Elliptic " } {TEXT 35 4 "K(m)" } {TEXT -1 57 " function shown in A&S is numerically equal to the Maple " } {TEXT 35 20 "EllipticK( sqrt(m) )" } {TEXT -1 9 " command." } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "FunctionAdvisor(definition, EllipticNome(k) )[1];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%-EllipticNomeG6#%\"kG-%$expG6#,$*(%#PiG\"\"\"-%+EllipticCKG F&F.-%*EllipticKGF&!\"\"F3 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf( eval(%, k=1/2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"+,(Qsz\"!#6F$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "EllipticModulus(EllipticNome(k)) = k; # valid for Re(k) > 0" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%0EllipticModulusG6#-%-EllipticNomeG6#%\"kGF* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf( eval(%, k=2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"\"#\"\"!F$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "EllipticNome(EllipticModulus(q)) = q; # valid for abs(q) < 1" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%-EllipticNomeG6#-%0EllipticModulusG6#%\"qGF* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf( eval(%, q=1/2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"+++++]!#5F$ "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "EllipticModulus" 2 "EllipticModulus" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Elliptic integrals" 2 "EllipticF" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "FunctionAdvisor" 2 "FunctionAdvisor" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "InverseJacobiPQ functions" 2 "InverseJacobiAM" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobiPQ functions" 2 "JacobiAM" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Jacobi Theta functions" 2 "JacobiTheta1" "" }{TEXT -1 1 " " } } }} cModulusG6#-%-EllipticNomeG6#%\"kGF* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf( eval(%, k=2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"\"#\"\"!F$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "EllipticNome(EllipticModulus(q)) = q; # valid for abs(q) < 1" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/-%-EllipticNomeG6#-%0EllipticModulusG6#%\"qGF* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf( eval(%, q=1/2) );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/$\"+++++]!#5F$ "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "EllipticModulus" 2 "EllipticModulus" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Elliptic integrals" 2 "EllipticF" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "FunctionAdvisor" 2 "FunctionAdvisor" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "InverseJacobiPQ functions" 2 "InverseJacobiAM" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobiPQ functions" 2 "JacobiAM" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Jacobi Theta functions" 2 "Jacop@{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 53 "Numeric Computations: Events, Status Flags, and Traps" } } {PARA 14 "> " 0 "" {TEXT -1 259 "Maple's numeric computation environment provides direct support for handling numeric events. There are six possible types of events, each described below, which are signaled when detected. When an event is signaled, it can be controlled in the following ways." } } {PARA 14 "> " 0 "" {TEXT -1 9 " * A " } {TEXT 22 4 "trap" } {TEXT -1 177 " may be installed for the event, indicating a routine to be invoked to determine the action to be taken. For information on how events are signaled and traps are installed, see " } {HYPERLNK 17 "NumericEventHandler" 2 "NumericEventHandler" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "NumericEventLocation" 2 "NumericEventLocation" "" }{TEXT -1 2 ". " } } {PARA 14 "> " 0 "" {TEXT -1 9 " * Ps]/{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 18 "plots[contourplot]" } {TEXT 30 22 " - 2D contour plotting" } } {PARA 3 "> " 0 "" {TEXT -1 20 "plots[contourplot3d]" } {TEXT 30 22 " - 3D contour plotting" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 17 " contourplot(" } {TEXT 35 5 "expr1" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 1 "=" } {TEXT 35 1 "a" } {TEXT -1 2 ".." } {TEXT 35 1 "b" } {TEXT -1 2 ", " } {TEXT 35 1 "y" } {TEXT -1 1 "=" } {TEXT 35 1 "c" } {TEXT -1 2 ".." } {TEXT 35 1 "d" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 17 " contourplot(" } {TEXT 35 1 "f" } {TEXT -1 2 ", " } {TEXT 35 1 "a" } {TEXT -1 2 ".." } {TEXT 35 1 "b" } {TEXT -1 2 ", " } {TEXT 35 1 "c" } {TEXT -1 2 ".." } {TEXT 35 1 "d" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 18 " contourplot([" } {TEXT 35 5 "exprf" } {TEX% {VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 21 "geometry[EulerCircle]" } {TEXT 30 44 " - find the Euler circle of a given triangle" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 17 " EulerCircle(" } {TEXT 35 3 "Elc" } {TEXT -1 2 ", " } {TEXT 35 1 "T" } {TEXT -1 3 ", '" } {TEXT 35 10 "centername" } {TEXT -1 2 "'=" } {TEXT 35 2 "cn" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 20 "T - " } {TEXT -1 9 "triangle " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 20 "Elc - " } {TEXT -1 29 "the name of the Euler circle " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 20 "'centername' = cn - " } {TEXT -1 17 "(optional) where " } {TEXT 35 2 "cn" } {TEXT -1 48 " is a name of the center of the Euler's circle. " } } /!on (the sine of the amplitude)" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 5 "nu - " } {TEXT -1 41 "algebraic expression (the characteristic)" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 5 "k - " } {TEXT -1 36 "algebraic expression (the parameter)" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 33 "The incomplete elliptic integral " } {TEXT 35 10 "EllipticPi" } {TEXT -1 15 " is defined by " } } {PARA 0 "> " 0 "" {TEXT 23 85 " EllipticPi(z,nu,k)\n = int(1/(1-nu*t^2)/sqrt(1-t^2)/sqrt(1-k^2*t^2),t=0..z)" } } {PARA 15 "> " 0 "" {TEXT -1 32 "The complete elliptic integrals " } {TEXT 35 10 "EllipticPi" } {TEXT -1 5 " and " } {TEXT 35 11 "EllipticCPi" } {TEXT -1 16 " are defined by " } } {PARA 0 "> " 0 "" {TEXT 23 96 " EllipticPi(nu,k) = EllipticPi(1,nu,k)\n EllipticCPi(nu,k) = EllipticPi(1,nu,sqrt(1-k^2))" } } {PARA 15 "> " 0 "" {TEXT -1 168 "Reference: Chapter 17, \"Elliptic Integrals\" of M. Abramowitz and I. Stegun, ``Handbook of Mathematical Functions.'' Dover Publications Inc., New York, 1046 p., (1965). " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "EllipticPi(0.1,0.2,0.3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+)Q%\\-5!#5 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "EllipticPi(0.2,0.3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+Pt@+=!\"* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "EllipticCPi(0.2,0.3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+&y??.$!\"* "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "EllipticF" 2 "EllipticF" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticK" 2 "EllipticK" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCK" 2 "EllipticCK" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticE" 2 "EllipticE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCE" 2 "EllipticCE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobiSN" 2 "JacobiSN" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "WeierstrassP" 2 "WeierstrassP" "" }{TEXT -1 1 " " } } }} ctions.'' Dover Publications Inc., New York, 1046 p., (1965). " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "EllipticPi(0.1,0.2,0.3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+)Q%\\-5!#5 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "EllipticPi(0.2,0.3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+Pt@+=!\"* "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "EllipticCPi(0.2,0.3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+&y??.$!\"* "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "EllipticF" 2 "EllipticF" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticK" 2 "EllipticK" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCK" 2 "EllipticCK" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticE" 2 "EllipticE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "EllipticCE" 2 "EllipticCE" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobiSN" 2 "JacobiSN" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "WeierstrassP" 2 "WeierstrassP" "" }{TEiinvalidpreprocerrorthereforbelowsecondxgfygfgftgsqrtexpandfalsegalphagfxgtruegalphagalphaaliavariablconsiderobjectotherproceedcomputatbeforrecursivecontainminimaltheirmoduloreducresultoccurrargumentrationalgeneratradicalrootofinsidappeardosetdomainmeantpropertquotiassignargumnotatwithrepresentmaynumberfunctionmustcoefficientotherwisfalstruereturnnameoptionalfunctnumbforminertfieldextensalgebraicoverpolynomialdivisdividevalaironmenvvarericeventhandlersgnormalizergprintlevelgordergmodgnewtablegdigitsgansgroundinggusehardwarefloatsgtestzerogbargbarfoogenvygfooenvydefinuserexamplresetincreasduringprintdeepwithinmodifishowprotectinsidonceonlytopatnoteactivcurrentsequencanamfunctionusingdoneeverprincipldiviscoefficiwhethdeterminzerotesttrivialrelativeendargsevalbprocindicatlevelupexecutstatementdisplacaussettoutputrepresdoesmaplperformcalculatrepresentdivisortermleadnormalizserikernelmodpinitialsetoperatpointfloatsignificantnumbcurrcomputatthirdsecondlastresultconsiderwithbeginnnameroundprintlevelordernumericeventhandlermodnewtablusehardwarefloatdigitpredefinlistfollowhappenreturnbeforthemrestorneednochangthesfunctwordothersupersedlocalunlescallsubprocedurallavailablvaluexitundonautomaticalbodyprocedurassignmsimplusedcanmiscindexvariablfluidenvzerotesttestzeronormalenvironmentvariablenvintegral_basis,algcurves5 integral_basis,numtheory)Zintegral_form,FunctionAdvisorintegral_transform,dsolveintegral_transforms,dsolveintegralform,FunctionAdvisorintegrand,student$integrate,VectorCalculuslintegrate_sols,DEtoolsBintegratesols,LinearOperatorsdintegratesols,linearoperatorsdintegrating_factor,DEtoolsЃintegrating_factors,DEtoolsЃintegrating_factors,dsolve|$integrating_factors_for_LODES,dsolve$integrating_factors_for_LODEs,dsolvei*{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 20 "inttrans[invfourier]" } {TEXT 30 28 " - inverse Fourier transform" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 16 " invfourier(" } {TEXT 35 4 "expr" } {TEXT -1 2 ", " } {TEXT 35 1 "w" } {TEXT -1 2 ", " } {TEXT 35 1 "t" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "expr - " } {TEXT -1 79 "expression, equation, or set of equations and/or expressions to be transformed " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "w - " } {TEXT -1 9 "variable " } {TEXT 35 4 "expr" } {TEXT -1 32 " is transformed with respect to " } {TEXT 35 1 "w" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "t - " } {TEXT -1 23 "parameter of transform " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 *LcontariowesternuniversitthesiphsolutbenghorbalscientistmathematicalcollectreferencshowirgadditionalalphaassumboundupperleastknowneedtheymaindealabilithavecfwdnhwpkjndcwyfqetfemkwlddcefdfexplnogcqjfdjcscgygparametsinblocksomeangerjgpigcosinerhsabovequalrecallasumcomputatnotexgfngfpochhammergvalgevalvalatmapltakeexampleliminatnormalbecausincreasaccuratlessbecomapproximathowevthsymbolicnegataccuracpointspecifparametermonomialfunctcasesimplifimessagerrorformreturnpowerintoexpandnonanyattemptnumericaldeducroutinallowaboutmadeassumptunlesunspecifisymbolacceptwillviaequationdifferentialprobleminitialworksuitablmakepropertdoeswhilimitatzeroconstantmapsthusimplementaroundwayotherperformaspectfollowdifferdefinliouvillriemanreservsometimtermliteratursenslimitonlyvalidvaluhandltgfintgnggammagnugxgfgdiffgintgammadiffresultintegraliteratformulastanATH 20 "6#7$%!GF$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(%);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"\"# "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "[1,a,3];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7%\"\"\"%!G\"\"$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "[1,b,3];" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"$ "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "names" 2 "names" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "NULL" 2 "NULL" "" }{TEXT -1 2 " " } } }} , while the empty symbol can. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a := ``;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"aG%!G "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "b := NULL;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"bG6\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c := ``;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"cG%!G "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "[a,b,c];" } } {PARA 11 "> " 1 "" {XPPMsession_.5AApBtGH,K_sk{ RQ-ݩG{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 14 "evalhf/hfarray" } {TEXT 30 23 " - handling of hfarrays" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT 35 6 "evalhf" } {TEXT -1 49 " handles only two special structures: arrays and " } {HYPERLNK 17 "rtables" 2 "rtables" "" }{TEXT -1 98 ". This help page describes rtables with their datatype option set to float[8] and complex[8]. The " } {HYPERLNK 17 "array" 2 "array" "" }{TEXT -1 15 " structure and " } {HYPERLNK 17 "rtable" 2 "rtable" "" }{TEXT -1 36 " structure (which is a supertype of " } {HYPERLNK 17 "Array" 2 "Array" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Matrix" 2 "Matrix" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Vector" 2 "Vector" "" }{TEXT -1 6 ", and " } {HYPERLNK 17 "hfarray" 2 "hfarray" "" }{TEXT -1 25 ") are also described in ?" } {HYPERLNK 17 "evalhf[array]" 2 "eva&G {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(Pi,2*Pi);" } } {PARA 2 "> " 1 "" {TEXT -1 87 "Error, invalid input: 2 arguments passed to f but only 1\npositional parameter specified" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "g := proc(a, b:=0, {c::polynom(integer,a):=a^2}, $)\n if b > 0 then\n a*b+c;\n else\n a+3*c;\n end if;\nend proc;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"gGt+6&%\"aG>%\"bG\"\"!%#~$G>'%\"cG-%(polynomG6$%(integerG9$* $F3\"\"#6\"F6F6@%2F*9%,&*&F3\"\"\"F9F " 0 "" {MPLTEXT 1 0 8 "g(x, 1);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*$F$\"\"#F% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g(y, c=2*y);" } } {PARA 2 "> " 1 "" {TEXT -1 81 "Error, (in g) cannot determine if this expression is true or\nfalse: 0 < (c = 2*y)" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g(z^4, 1, 2*z);" } } {PARA 2 "> " 1 "" {TEXT -1 88 "Error, invalid input: 3 arguments passed to g but only 2\npositional parameters specified" } } } } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "Procedures" 2 "procedure" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "paramprocessing" 2 "paramprocessing" "" }{TEXT -1 2 " " } } }} er,a):=a^2}, $)\n if b > 0 then\n a*b+c;\n else\n a+3*c;\n end if;\nend proc;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"gGt+6&%\"aG>%\"bG\"\"!%#~$G>'%\"cG-%(polynomG6$%(integerG9$* $F3\"\"#6\"F6F6@%2F*9%,&*&F3\"\"\"F9F " 0 "" {MPLTEXT 1 0 8 "g(x, 1);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*$F$\"\"#F% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g(y, c=2*y);" } } {PARA 2 "> " 1 "" {TEXT -1 81 "Error, (in g) cannot determine if this expression is true or\nfalse: 0 < (c = 2*y)" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g(z^4, 1, 2*z);" } } {PARA 2 "> " 1 "" {TEXT -1 88 "Error, invalid input: 3 arguments passed to g but only 2\npo$MathematicalFunctions,SearchFunction9find Maple routines related to a mathematical functionMathematicalFunctions,Series50compute series expansion for mathematical functionMathematicalFunctions,get+.return mathematical function information$MathematicalFunctions,searchfunction9find Maple routines related to a mathematical functionMathematicalFunctions,series50compute series expansion for mathematical functionMathieu7=EDescription of transitive groups_Mathieu functionstLinearAlgebra,NormalizeLinearAlgebra:-NormalizeLinearAlgebra,LinearAlgebra:-NormalizeNormalizenormalizeLinearAlgebra,normalizetLinearAlgebra,MatrixAddLinearAlgebra:-MatrixAddLinearAlgebra,LinearAlgebra:-MatrixAddLinearAlgebra:-VectorAddLinearAlgebra,LinearAlgebra:-VectorAddVectorAddLinearAlgebra,VectorAddMatrixAddLinearAlgebra,MatrixAddtDEtools,hamilton_eqstverify,subsettconvert,explnt.DEtools,de2diffopdiffop2deDEtools,diffop2deܲirreduciblpretendsubstitutdetectpreprocwellgeneratdoesreduciblzgrootofgxgxgftgfygagygftgfgsqrtconsiderobjectproceedcomputatbeforcheckindependencskipbasionlyexpresssatisfineedconditerrorotherwismustexpandoutputpossiblfarfactorizatpartialpreservallspecifigivenminimaltheirmoduloreducoccurrchanganothsessvaryvariablorderanycontaintheynumeratorleadfollownormalpropertsatisfargumvalunotealsoquantitformalviewwhosbelowargumentfactorobtainproductwordothercoefficinormalizatuseparticularconstantmultiplicatdefincorrespondelemprimrelativeradicalrootofinsidappeardoindeterminatemptnondenotresultsamereturnrationalprecisemoreintegerringtypicalwillaccordingproceguesreasonablmakeattemptspecifmaplwaynosincfraccaseassigndenominatoroutclearfractcandefinitabovunitchoicupuniqusuchassociatdivisorcommongreatestdomaingcdspeakgeneralnotatrepresentmaynumbercoefficientwithconteCLinearAlgebra,basis$return a basis for a vector spaceLinearAlgebra,bezout1kconstruct the Bezout Matrix of two polynomialsLinearAlgebra,bilinearBpcompute the dot product (standard inner product) of two VectorsLinearAlgebra,bilinearformBpcompute the dot product (standard inner product) of two VectorsLinearAlgebra,characteristic\construct the characteristic Matrixconstruct the characteristic polynomial of a Matrix&LinearAlgebra,characteristicpolynomial6construct the characteristic polynomial of a Matrix \nprintlevel Rounding Testzero \n " } } {PARA 0 "> " 0 "" {TEXT -1 1 " " } } {PARA 15 "> " 0 "" {TEXT -1 32 "A variable name beginning with \"" } {TEXT 35 4 "_Env" } {TEXT -1 41 "\" is considered an environment variable. " } } {PARA 15 "> " 0 "" {TEXT 35 1 "%" } {TEXT -1 2 ", " } {TEXT 35 2 "%%" } {TEXT -1 6 ", and " } {TEXT 35 3 "%%%" } {TEXT -1 98 " are the results of the last, second last, and third last computation in the current environment. " } } {PARA 15 "> " 0 "" {TEXT 35 6 "Digits" } {TEXT -1 101 " is the number of significant digits used for floating-point operations, and is set to 10 initially. " } } {PARA 15 "> " 0 "" {TEXT 35 3 "mod" } {TEXT -1 11 " is set to " } {TEXT 35 6 "`modp`" } {TEXT -1 12 " initially. " } } {PARA 15 "> " 0 "" {TEXT 35 10 "Normalizer" } {TEXT -1 11 " is set to " } {TEXT 35 8 "`normal`" } {TEXT -1 88 " initially, and is used by the kernel in series to normalize leading terms of divisors. " } } {PARA 15 "> " 0 "" {TEXT 35 5 "Order" } {TEXT -1 119 " represents the order of series calculations performed by Maple. It does not represent the order of the series output. " } {TEXT 35 5 "Order" } {TEXT -1 24 " is set to 6 initially. " } } {PARA 15 "> " 0 "" {TEXT -1 15 "The setting of " } {TEXT 35 10 "printlevel" } {TEXT -1 91 " causes the display of the results of all statements executed up to the level indicated by " } {TEXT 35 10 "printlevel" } {TEXT -1 13 ". Initially, " } {TEXT 35 10 "printlevel" } {TEXT -1 14 " is set to 1. " } } {PARA 15 "> " 0 "" {TEXT 35 8 "Testzero" } {TEXT -1 21 " is initialized with " } {TEXT 35 41 "proc() evalb(Normalizer(args[1]) = 0) end" } {TEXT -1 47 ", which is a relatively trivial test for zero. " } {TEXT 35 8 "Testzero" } {TEXT -1 210 " is used in the kernel by series, when it needs to determine whether a leading coefficient is zero for division. In principle, every test for zero and every call to normal should be done using these functions. " } } {PARA 15 "> " 0 "" {TEXT -1 9 "The call " } {TEXT 35 21 "anames('environment')" } {TEXT -1 84 " returns a sequence of all the currently active environment variables. Note that an " } {TEXT 35 4 "_Env" } {TEXT -1 195 " variable is not considered to be an environment variable at the top level; only once execution is inside a procedure is an environment set up to protect the top-level value of the variable. So, " } {TEXT 35 4 "_Env" } {TEXT -1 80 " variables only show up if these variables are modified within a procedure, and " } {TEXT 35 21 "anames('environment')" } {TEXT -1 72 " is called from within the procedure or at a deeper level of execution. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "t := proc() Digits := Digits + 4; end proc:\nprint(Digits);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"#5 "}}} {PARA 0 "> " 0 "" {TEXT -1 25 "During the execution of \"" } {TEXT 35 1 "t" } {TEXT -1 3 "\", " } {TEXT 35 6 "Digits" } {TEXT -1 58 " is increased by 4, and automatically reset on exit from \"" } {TEXT 35 1 "t" } {TEXT -1 3 "\". " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "t();" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"#9 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "print(Digits);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"#5 "}}} {PARA 0 "> " 0 "" {TEXT -1 68 "The following is an example of a user defined environment variable. " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "_Envy := foo;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%&_EnvyG%$fooG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "p := proc() _Envy := bar end proc:\np();" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%$barG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "_Envy;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%$fooG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "anames(environment);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "60%)TestzeroG%2UseHardwareFloatsG%)RoundingG%\"%G%%_ansG%$%%% G%'DigitsG%/index/newtableG%$modG%#%%G%&OrderG%+printlevelG% +NormalizerG%5NumericEventHandlersG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "p := proc() _Envy := bar; anames(environment) end proc:\np();" } } {PARA 11 "> " 1 "" {XPPMATH 20 "61%)TestzeroG%2UseHardwareFloatsG%)RoundingG%\"%G%%_ansG%$%%% G%'DigitsG%/index/newtableG%$modG%#%%G%&OrderG%+printlevelG% +NormalizerG%5NumericEventHandlersG%&_EnvyG "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "Digits" 2 "Digits" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "ditto" 2 "ditto" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "mod" 2 "mod" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "normal" 2 "normal" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "NumericEventHandler" 2 "NumericEventHandler" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Order" 2 "Order" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "printlevel" 2 "printlevel" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Rounding" 2 "Rounding" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "series" 2 "series" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "UseHardwareFloats" 2 "UseHardwareFloats" "" }{TEXT -1 2 " " } } }} X {VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 22 "FileTools[IsDirectory]" } {TEXT 30 37 " - determine if a file is a directory" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 17 " IsDirectory(" } {TEXT 35 4 "file" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "file - " } {TEXT -1 47 "filename or the file descriptor of an open file" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 17 "IsDirectory(file)" } {TEXT -1 17 " command returns " } {TEXT 35 4 "true" } {TEXT -1 4 " if " } {TEXT 35 4 "file" } {TEXT -1 50 " refers to a directory. If it is not a directory, " } {TEXT 35 5 "false" } {TEXT -1 14 " is returned. " } } {PARA 15 "> " 0 "" {TEXT -1 92 "If the file does not exist or an inE-O{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 21 "Numerical Integration" } } {PARA 4 "> " 0 "usage" {TEXT -1 17 "Calling Sequences" } } {PARA 0 "> " 0 "" {TEXT -1 15 " evalf(Int(" } {TEXT 35 1 "f" } {TEXT -1 4 ", x=" } {TEXT 35 4 "a..b" } {TEXT -1 7 ", ...))" } } {PARA 0 "> " 0 "" {TEXT -1 15 " evalf(Int(" } {TEXT 35 1 "f" } {TEXT -1 2 ", " } {TEXT 35 4 "a..b" } {TEXT -1 7 ", ...))" } } {PARA 0 "> " 0 "" {TEXT -1 15 " evalf(Int(" } {TEXT 35 1 "f" } {TEXT -1 26 ", list-of-equations, ...))" } } {PARA 0 "> " 0 "" {TEXT -1 15 " evalf(Int(" } {TEXT 35 1 "f" } {TEXT -1 23 ", list-of-ranges, ...))" } } {PARA 0 "> " 0 "" {TEXT -1 15 " evalf(int(" } {TEXT 35 1 "f" } {TEXT -1 4 ", x=" } {TEXT 35 4 "a..b" } {TEXT -1 2 "))" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 20 "f - " } {TEXnumeric,rosenbrock"/numerical solution of stiff ODE numericEvent+triggers a numeric event numeric_BVP @numerical solution of ODE BVP numeric_DAE$numerical solution of ODE DAE IVP numeric_IVP ]numerical solution of ODE IVP numeric_bvp @numerical solution of ODE BVP numeric_dae$numerical solution of ODE DAE IVPnumeric_eventsp numeric_ivp ]numerical solution of ODE IVP numeric_modelwnumeric_overvieww numeric_typeknumeric types in Maple numerical+mevaluate using floating-point arithmetic numericevent+triggers a numeric event evalf,details+1:evaluate using floating-point arithmeticevalf,evalfIntnumerical integrationevalf,evalfSum(numerical summationevalf,evalfintnumerical integrationevalf,evalfsum(numerical summationevalf,hypergeom+mevaluate using floating-point arithmetic evalf,intnumerical integration evalf,sum(numerical summationevalfIntnumerical integrationevalfSum(numerical summationevalfintnumerical integrationevalfsum(numerical summationnumarg ҽsK{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 12 "ImportMatrix" } {TEXT 30 30 " - import a Matrix from a file" } } {PARA 3 "> " 0 "" {TEXT -1 12 "ExportMatrix" } {TEXT 30 28 " - export a Matrix to a file" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 18 " ImportMatrix(" } {TEXT 35 1 "f" } {TEXT -1 2 ", " } {TEXT 35 1 "s" } {TEXT -1 2 ", " } {TEXT 35 2 "fo" } {TEXT -1 2 ", " } {TEXT 35 1 "d" } {TEXT -1 2 ", " } {TEXT 35 2 "de" } {TEXT -1 2 ", " } {TEXT 35 2 "tr" } {TEXT -1 2 ", " } {TEXT 35 2 "sl" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 18 " ExportMatrix(" } {TEXT 35 1 "f" } {TEXT -1 2 ", " } {TEXT 35 1 "M" } {TEXT -1 2 ", " } {TEXT 35 1 "t" } {TEXT -1 2 ", " } {TEXT 35 2 "de" } {TEXT -1 2 ", " } {TEXT 35 2 "tr" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -(o {VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 22 "ListTools[FlattenOnce]" } {TEXT 30 40 " - flatten a (possible nested) list once" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 17 " FlattenOnce(" } {TEXT 35 1 "L" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 4 "L - " } {TEXT -1 4 "list" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 14 "FlattenOnce(L)" } {TEXT -1 39 " function converts any element in list " } {TEXT 35 1 "L" } {TEXT -1 19 " which are of type " } {TEXT 35 4 "list" } {TEXT -1 26 " to expression sequences. " } } {PARA 15 "> " 0 "" {TEXT -1 26 "This procedure has option " } {TEXT 35 6 "inline" } {TEXT -1 89 "; therefore, it can be used within a procedure without the ov6 {TEXT -1 85 " into a list of equations, where the left-hand sides of the equations are taken from " } {TEXT 35 1 "u" } {TEXT -1 63 " and the right-hand sides from the corresponding components of " } {TEXT 35 1 "v" } {TEXT -1 2 ". " } } {PARA 14 "> " 0 "" {TEXT -1 23 "The two input objects, " } {TEXT 35 1 "u" } {TEXT -1 5 " and " } {TEXT 35 1 "v" } {TEXT -1 61 ", must have the same shapes and sizes, and if only one is an " } {HYPERLNK 17 "array" 2 "array" "" }{TEXT -1 4 " or " } {HYPERLNK 17 "Array" 2 "Array" "" }{TEXT -1 35 " then its indices must start at 1. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Equate( [1,2], );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7$/\"\"\"%\"aG/\"\"#%\"bG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Equate( <<1,2>|<3,4>|<5,6>>, [[a,b,c],[d,e,f]]);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7(/\"\"\"%\"aG/\"\"$%\"bG/\"\"&%\"cG/\"\"#%\"dG/\"\"%%\"eG/\"\"'%\"fG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Equate( <1,2,3>, );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#7%/\"\"\"%\"aG/\"\"#%\"bG/\"\"$%\"cG "}}} {PARA 0 "> " 0 "" {TEXT -1 53 "This is invalid because the dimensions do not match. " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Equate( [[1,2],[3,4]], array(1..4,[a,b,c,d]) );" } } {PARA 2 "> " 1 "" {TEXT -1 49 "Error, (in Equate) unable to equate these objects" } } } {PARA 0 "> " 0 "" {TEXT -1 56 "This is invalid because the index origins do not match. " } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Equate( Array(-2..1,[1,2,3,4]), array(1..4,[a,b,c,d]) );" } } {PARA 2 "> " 1 "" {TEXT -1 49 "Error, (in Equate) unable to equate these objects" } } } } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "Array" 2 "Array" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "convert" 2 "convert" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Matrix" 2 "Matrix" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Vector" 2 "Vector" "" }{TEXT -1 2 " " } } }} "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Equ{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 14 "group[invperm]" } {TEXT 30 32 " - find inverse of a permutation" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 13 " invperm(" } {TEXT 35 4 "perm" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "perm - " } {TEXT -1 38 "permutation in disjoint cycle notation" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 78 "The inverse is computed, and the result is output in disjoint cycle notation. " } } {PARA 15 "> " 0 "" {TEXT -1 87 "The command with(group,invperm) allows the use of the abbreviated form of this command." } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(group):\ninvperm([[4,6,5], [2,a&` {VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 5 "evaln" } {TEXT 30 21 " - evaluate to a name" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 11 " evaln(" } {TEXT 35 4 "expr" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "expr - " } {TEXT -1 10 "expression" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 5 "evaln" } {TEXT -1 153 " function is used to ``evaluate to a name'' or, more generally, to form an assignable object. The name (assignable object) will not itself be evaluated. " } } {PARA 15 "> " 0 "" {TEXT -1 38 "A typical example of its use would be " } {TEXT 35 23 "divide(a,b,evaln(t[i]))" } {TEXT -1 7 " where " } {TEXT 35 1 "i" } {TEXT -1 19 " is the index of a " } {TEXT 35 ?#{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 5 "evalb" } {TEXT 30 35 " - evaluate as a Boolean expression" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 11 " evalb(" } {TEXT 35 1 "x" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 4 "x - " } {TEXT -1 11 "expression " } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 5 "evalb" } {TEXT -1 135 " command forces the evaluation of expressions involving relational operators, using a three-valued logic system. The return values are " } {TEXT 35 4 "true" } {TEXT -1 2 ", " } {TEXT 35 5 "false" } {TEXT -1 6 ", and " } {TEXT 35 4 "FAIL" } {TEXT -1 72 ". If evaluation is not possible, an unevaluated expression is returned. " } } {PARA 15 "> " 0 ""mples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "e := a = b;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"eG/%\"aG%\"bG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "type( e, 'equation' );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%%trueG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "e := f(x) < g(x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%\"eG2-%\"fG6#%\"xG-%\"gGF( "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "type( e, 'equation' );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%&falseG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "type( e, `<` );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%%trueG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "lhs( e );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#-%\"fG6#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rhs( e );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#-%\"gG6#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqs := { a1 = b1, a2 = b2, a3 = b3 };" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#>%$eqsG<%/%#a1G%#b1G/%#a2G%#b2G/%#a3G%#b3G "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map( lhs, eqs );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<%%#a1G%#a2G%#a3G "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a = 2;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#/%\"aG\"\"# "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%\"aG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assign( %% );\na;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"\"# "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "assign" 2 "assign" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "boolean" 2 "boolean" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "evalb" 2 "evalb" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "lhs" 2 "lhs" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "op" 2 "op" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "plots[inequal]" 2 "plots[inequal]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "rhs" 2 "rhs" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "type" 2 "type" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "type[equation]" 2 "type[equation]" "" }{TEXT -1 2 " " } } }} "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 ven with this list as ranking, the equations in each returned system are of the form " } {TEXT 35 78 "[..f(x)..=..(f(x), g(x), h(x)).., ..g(x)=..(g(x), h(x)).., ..h(x)..=..(h(x)) ]" } {TEXT -1 38 ". So, the last equation involves only " } {TEXT 35 4 "h(x)" } {TEXT -1 41 ", the next to the last equation involves " } {TEXT 35 4 "g(x)" } {TEXT -1 5 " and " } {TEXT 35 4 "h(x)" } {TEXT -1 70 ", and the first equation involves the three functions. This is called " } {TEXT 22 21 "sequential uncoupling" } {TEXT -1 1 "." } } {PARA 15 "> " 0 "" {TEXT -1 319 "In the case of a system of partial differential equations (PDEs), the situation is the same as with ODEs but the concept of sequential uncoupling requires clarification. Concretely, instead of one equation related to each function, one could receive many equations. For instance, if a system depending on the functions " } {TEXT 35 30 "[f(x,y,z), g(x,y,z), h(x,y,z)]" } {TEXT -1 95 " is given and with this list as ranking, the equations in each returned system areO 1 "p" } {TEXT -1 34 " is supplied and the test returns " } {TEXT 35 5 "false" } {TEXT -1 34 ", then a valuation is assigned to " } {TEXT 35 1 "p" } {TEXT -1 50 " which demonstrates a negative result. Otherwise, " } {TEXT 35 1 "p" } {TEXT -1 13 " is assigned " } {TEXT 35 4 "NULL" } {TEXT -1 2 ". " } } {PARA 14 "> " 0 "" {TEXT 37 5 "Note:" } {TEXT -1 41 " The test may be significantly faster if " } {TEXT 35 1 "p" } {TEXT -1 15 " is not given. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(Logic):\nEquivalent(a &and (a &or b), a);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%%trueG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Equivalent(a &iff (a &or b), b &implies a);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%%trueG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Equivalent(a &implies b, b &implies a, 'p');" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%&falseG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "p;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<$/%\"6<{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 3 "erf" } {TEXT 30 21 " - The Error Function" } } {PARA 3 "> " 0 "" {TEXT -1 4 "erfc" } {TEXT 30 62 " - The Complementary Error Function and its Iterated Integrals" } } {PARA 3 "> " 0 "" {TEXT -1 4 "erfi" } {TEXT 30 31 " - The Imaginary Error Function" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 9 " erf(" } {TEXT 35 1 "x" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 10 " erfc(" } {TEXT 35 1 "x" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 10 " erfc(" } {TEXT 35 1 "n" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 10 " erfi(" } {TEXT 35 1 "x" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 4 "x - " } {TEXT -1 20 "algebraic expression" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 4 "n - " } {TEXT -1 55 "algebraic expression, understood to be an integer >= -1" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 46 "The error function is defined for all complex " } {TEXT 35 1 "x" } {TEXT -1 4 " by " } } {PARA 0 "> " 0 "" {TEXT 23 50 " erf(x) = 2/sqrt(Pi) * int(exp(-t^2), t=0..x)" } } {PARA 15 "> " 0 "" {TEXT -1 47 "The complementary error function is defined by " } } {PARA 0 "> " 0 "" {TEXT 23 68 " erfc(x) = 1 - erf(x) = 1 - 2/Pi^(1/2) * int(exp(-t^2), t=0..x)" } } {PARA 15 "> " 0 "" {TEXT -1 74 "The iterated integrals of the complementary error function are defined by " } } {PARA 0 "> " 0 "" {TEXT 23 103 " erfc(-1, x) = 2/sqrt(Pi) * exp(-x^2)\n erfc(n, x) = int(erfc(n-1,t), t=x..infinity) n >= 0" } } {PARA 14 "> " 0 "" {TEXT -1 6 "(Note " } {TEXT 35 20 "erfc(0, x) = erfc(x)" } {TEXT -1 3 ".) " } } {PARA 15 "> " 0 "" {TEXT -1 43 "The imaginary error function is defined by " } } {PARA 0 "> " 0 "" {TEXT 23 64 " erfi(x) = -I*erf(I*x) = 2/sqrt(Pi) * int(exp(t^2), t=0..x)" } } {PARA 15 "> " 0 "" {TEXT -1 35 "All of these functions are entire. " } } {PARA 15 "> " 0 "" {TEXT -1 23 "Reference: A. Erdelyi, " } {TEXT 22 31 "Higher Transcendental Functions" } {TEXT -1 31 ", Volume 2, McGraw-Hill, 1953. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "erf(3);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#-%$erfG6#\"\"$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"+&4z(****!#5 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "erfc(3.);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#$\"++(\\!4A!#9 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "erf(1.-1.*I);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#^$$\"+#G^hJ\"!\"*$!+#pMX!>!#5 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "erfc(1.5-2.85*I);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#^$$!+*)[1#G'!\")$!+&\\nh0\"F& "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(erfc(5,x), x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,$-%%erfcG6$\"\"%%\"xG!\"\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "erfi(-x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,$-%%erfiG6#%\"xG!\"\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(erfi(x),x,4);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#+)%\"xG,$*$%#PiG#!\"\"\"\"#F*\"\"\",$F&#F*\"\"$F.-%\"OG6#F+\"\"% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(erfc(2,x),x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,,*$%\"xG\"\"##\"\"\"F&*&F%F&-%$erfG6#F%F(#!\"\"F&*(F%F(%#PiGF--%$ expG6#,$F$F.F(F-#F(\"\"%F(F*#F.F6 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "convert(%,erfc);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,**$%\"xG\"\"##\"\"\"F&*&F%F&,&F(F(-%%erfcG6#F%!\"\"F(#F.F&*(F%F(% #PiGF/-%$expG6#,$F$F.F(F/F+#F(\"\"% "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "Fresnel" 2 "Fresnel" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "dawson" 2 "dawson" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "convert" 2 "convert" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "inifcns" 2 "inifcns" "" }{TEXT -1 2 " " } } }} " "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "erfi(-x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,$-%%erfiG6#%\"xG!\"\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(erfi(x),x,4);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#+)%\"xG,$*$%#PiG#!\"\"\"\"#F*\"\"\",$F&#F*\"\"$F.-%\"OG6#F+\"\"% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(erfc(2,x),x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,,*$%\"xG\"\"##\"\"\"F&*&F%F&-%$erfG6#F%F(#!\"\"F&*(F%F(%#PiGF--%$ expG6#,$F$F.F(F-#F(\"\"%F(F*#F.F6 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "convert(%,erfc);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,**$%\"xG\"\"##\"\"\"F&*&F%F&,&F(F(-%%erfcG6#F%!\"\"F(#F.F&*(F%F(% #PiGF/-%$expG6#,$F$F.F(F/F+#F(\"\"% "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "Fresnel" 2 "Fresnel" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "dawson" 2 "dawson" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "convert" 2 "convert"InverseJacobiNC1InverseJacobi elliptic and amplitude functionsInverseJacobiND1InverseJacobi elliptic and amplitude functionsInverseJacobiNS1InverseJacobi elliptic and amplitude functionsInverseJacobiSC1InverseJacobi elliptic and amplitude functionsInverseJacobiSD1InverseJacobi elliptic and amplitude functionsInverseJacobiSN1InverseJacobi elliptic and amplitude functionsInverse_length+5units of inverse_lengthInvfuncinverse function tableIota.)generate string containing character rangesIrreduc inert irreducibility functionIs make and query assumptions1:ignameinstancaroundplacdirectalwayxxendlocalprocnewfunctinputnontimefollowmaplcodeshowclearsuchufuqlresultsubroutinfunctaddingzgrootofgsumgrootofftlmtanexpgintgintegralapproximatcomputlllllfxgsqrtztficosvksjpaeftjargumpasstheysubfunctcorrespondautomaticalmathematicknowninitialmostsectchecktypequotrightunevaluatrequirspecifinoteargumentinvokgeneralprocedurmustdefinnewinterfacextendsequenccallindexchangwithoutmaxdigitkerneloptexcedoesintegpositanyusesvaluassigndefaultvariablenvironmcomputatallcontrolcanyouprecisnumericcomplexreturnoutputsumsummatmoreintwithconjunctuseintegratperformerfcosharctansinlnfunctionmathematicalgammaexppiexamplconstantinvolvsubexpressnumericaldescriptcommandaboutcompletcontainpagehelpinformatbasicdigitnumbspecifoptionalexpressarithmeticpointfloatusingevaluatdetailevalft process (that is, variables are not shared). " } } {PARA 15 "> " 0 "" {TEXT -1 136 "The child process does not have access to the parent's output display. Therefore, printing from the child will not display any results. " } } {PARA 15 "> " 0 "" {TEXT -1 7 "If the " } {TEXT 35 4 "fork" } {TEXT -1 18 " operation fails, " } {TEXT 35 4 "fork" } {TEXT -1 21 " generates an error. " } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 4 "fork" } {TEXT -1 102 " function is only available on platforms that support true concurrent processing (for example, UNIX). " } } {PARA 15 "> " 0 "" {TEXT -1 21 "WARNING: Do not call " } {TEXT 35 4 "fork" } {TEXT -1 253 " at the top level (that is, at the prompt, or as a top level statement in a file being read). Doing so results in two Maple processes, both of which compete for subsequent input. Likewise, make sure that no forked process ever returns to the top level. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107~Vto " } {TEXT 35 9 "msgString" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 28 "expr1, expr2, ... - " } {TEXT -1 35 "expression sequence (possibly null)" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 5 "error" } {TEXT -1 280 " statement raises an exception. Execution of the current statement sequence is interrupted, and the block and procedure call stack is popped until either an exception handler is encountered, or an execution returns to the top level (in which case the exception becomes an error). " } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 9 "msgString" } {TEXT -1 273 " is a string value that is independent of any parameters that are to be part of the message (for instance, the string complaining about an unassigned variable must not mention the variable by name). The string can, however, contain numbered parameter locations of the form " } {TEXT 35 4 "'%n'" } {TEXT -1 6 " when " } {TEXT 35 1 "n" } {TEXT -1 16 " is an integer. " } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 25 "msgParam1, msgParam2, ..." } {TEXT -1 103 " are one or more arbitrary Maple objects that are substituted into numbered parameter locations in the " } {TEXT 35 9 "msgString" } {TEXT -1 65 " in the event that the exception is printed as an error message. " } } {PARA 15 "> " 0 "" {TEXT -1 6 "If no " } {TEXT 35 9 "msgString" } {TEXT -1 11 " is given, " } {TEXT 35 5 "error" } {TEXT -1 47 " raises the most recently occurring exception. " } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 5 "error" } {TEXT -1 135 " statement evaluates its arguments, and then creates an exception object, which is an expression sequence with the following elements. " } } {PARA 0 "> " 0 "" {TEXT 23 256 " - The name of the procedure in which the exception was raised\n (hereinafter referred to as the procName). If the exception\n was raised at the top level, the constant 0 appears here.\n - The msgString.\n - The msgParams, if any." } } {PARA 15 "> " 0 "" {TEXT -1 64 "The created exception object is assigned to the global variable " } {TEXT 35 13 "lastexception" } {TEXT -1 32 " as an expression sequence. The " } {TEXT 35 9 "msgString" } {TEXT -1 5 " and " } {TEXT 35 8 "msgParam" } {TEXT -1 30 " sequence is also assigned to " } {TEXT 35 9 "lasterror" } {TEXT -1 43 " as a symbol (for backward compatibility). " } } {PARA 15 "> " 0 "" {TEXT -1 7 "In the " } {TEXT 35 9 "msgString" } {TEXT -1 98 ", numbered parameters are used as placeholders for actual values or names. For example, the error " } {TEXT 35 43 "\"f has a 2nd argument, x, which is missing\"" } {TEXT -1 21 " is specified by the " } {TEXT 35 9 "msgString" } {TEXT -1 1 " " } {TEXT 35 45 "\"%1 has a %-2 argument, %3, which is missing\"" } {TEXT -1 10 ", and the " } {TEXT 35 8 "msgParam" } {TEXT -1 2 "s " } {TEXT 35 1 "f" } {TEXT -1 2 ", " } {TEXT 35 1 "2" } {TEXT -1 6 ", and " } {TEXT 35 1 "x" } {TEXT -1 2 ". " } } {PARA 15 "> " 0 "" {TEXT -1 57 "Each numbered parameter consists of the percent symbol, \"" } {TEXT 35 1 "%" } {TEXT -1 41 "\", optionally followed by a minus sign, \"" } {TEXT 35 1 "-" } {TEXT -1 56 "\", followed by one or more digits, making up an integer " } {TEXT 35 1 "n" } {TEXT -1 31 ". At message display time, the " } {TEXT 35 1 "n" } {TEXT -1 3 "th " } {TEXT 35 8 "msgParam" } {TEXT -1 45 " are substituted for the numbered parameter. " } } {PARA 15 "> " 0 "" {TEXT -1 33 "A numbered parameter of the form " } {TEXT 35 2 "%n" } {TEXT -1 14 " displays the " } {TEXT 35 1 "n" } {TEXT -1 3 "th " } {TEXT 35 8 "msgParam" } {TEXT -1 4 " in " } {HYPERLNK 17 "lprint" 2 "lprint" "" }{TEXT -1 11 " notation. " } } {PARA 15 "> " 0 "" {TEXT -1 33 "A numbered parameter of the form " } {TEXT 35 3 "%-n" } {TEXT -1 14 " displays the " } {TEXT 35 1 "n" } {TEXT -1 3 "th " } {TEXT 35 8 "msgParam" } {TEXT -1 73 ", assumed to be a Maple integer, in ordinal form. For example, the value " } {TEXT 35 1 "2" } {TEXT -1 18 " is displayed as \"" } {TEXT 35 3 "2nd" } {TEXT -1 3 "\". " } } {PARA 15 "> " 0 "" {TEXT -1 23 "The special parameter, " } {TEXT 35 2 "%0" } {TEXT -1 19 ", displays all the " } {TEXT 35 8 "msgParam" } {TEXT -1 35 "s, separated by a comma and space. " } } {PARA 15 "> " 0 "" {TEXT -1 12 "You can use " } {HYPERLNK 17 "StringTools[FormatMessage]" 2 "StringTools[FormatMessage]" "" }{TEXT -1 40 " to format the string produced from the " } {TEXT 35 13 "lastexception" } {TEXT -1 8 " value. " } } {PARA 15 "> " 0 "" {TEXT -1 27 "For consistency, the Maple " } {HYPERLNK 17 "WARNING" 2 "WARNING" "" }{TEXT -1 22 " function also uses a " } {TEXT 35 9 "msgString" } {TEXT -1 5 " and " } {TEXT 35 8 "msgParam" } {TEXT -1 3 "s. " } } {PARA 15 "> " 0 "" {TEXT -1 45 "Exceptions can be caught by using either the " } {HYPERLNK 17 "try" 2 "try" "" }{TEXT -1 3 "..." } {HYPERLNK 17 "catch" 2 "catch" "" }{TEXT -1 48 " facility or the obsolete, but still supported, " } {HYPERLNK 17 "traperror" 2 "traperror" "" }{TEXT -1 11 " function. " } } {PARA 15 "> " 0 "" {TEXT -1 81 "In Maple V Release 5.1 and earlier, this mechanism to raise an exception was the " } {TEXT 35 5 "ERROR" } {TEXT -1 74 " function. This facility is obsolete, but is still supported in Maple 10. " } } {PARA 15 "> " 0 "" {TEXT -1 14 "A call to the " } {TEXT 35 5 "ERROR" } {TEXT -1 61 " function is semantically equivalent to the execution of the " } {TEXT 35 5 "error" } {TEXT -1 50 " statement, with a few exceptions, outlined next. " } } {PARA 15 "> " 0 "" {TEXT -1 108 "If called with a first argument, that is, a string or a symbol containing a numbered positional parameters, " } {TEXT 35 5 "ERROR" } {TEXT -1 31 " works exactly the same way as " } {TEXT 35 5 "error" } {TEXT -1 22 ". The same is true if " } {TEXT 35 5 "ERROR" } {TEXT -1 64 " is called with a single argument that is a string or a symbol. " } } {PARA 15 "> " 0 "" {TEXT -1 117 "If called with multiple arguments, the first of which is a string or symbol, but without positional parameters, then " } {TEXT 35 12 "ERROR(a,b,c)" } {TEXT -1 18 " is equivalent to " } {TEXT 35 17 "error \"a, %0\",b,c" } {TEXT -1 48 ". In other words, the first parameter becomes a " } {TEXT 35 9 "msgString" } {TEXT -1 7 " with \"" } {TEXT 35 4 ", %0" } {TEXT -1 54 "\" appended to it, and the remaining parameters become " } {TEXT 35 8 "msgParam" } {TEXT -1 11 "s for this " } {TEXT 35 9 "msgString" } {TEXT -1 2 ". " } } {PARA 15 "> " 0 "" {TEXT -1 62 "If called with any number of arguments, the first of which is " } {TEXT 22 3 "not" } {TEXT -1 26 " a string or symbol, then " } {TEXT 35 12 "ERROR(a,b,c)" } {TEXT -1 18 " is equivalent to " } {TEXT 35 16 "error \"%0\",a,b,c" } {TEXT -1 22 ". In other words, the " } {TEXT 35 9 "msgString" } {TEXT -1 1 " " } {TEXT 35 4 "\"%0\"" } {TEXT -1 42 " is implied, and all the arguments become " } {TEXT 35 8 "msgParam" } {TEXT -1 3 "s. " } } {PARA 15 "> " 0 "" {TEXT -1 14 "In all cases, " } {TEXT 35 9 "lasterror" } {TEXT -1 46 " is assigned exactly the parameters passed to " } {TEXT 35 5 "ERROR" } {TEXT -1 6 ", and " } {TEXT 35 13 "lastexception" } {TEXT -1 81 " is assigned as though the equivalent (as described in the preceding paragraphs) " } {TEXT 35 5 "error" } {TEXT -1 30 " statement had been executed. " } } {PARA 15 "> " 0 "" {TEXT -1 17 "The semantics of " } {TEXT 35 5 "error" } {TEXT -1 76 " have been carefully designed to remain backward compatible with the use of " } {TEXT 35 5 "ERROR" } {TEXT -1 37 " in Maple V Release 5.1 and earlier. " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f := proc (x) if x<0 then error \"invalid x: %1\", x else x^(1/2) end if end proc:\nf(-3);" } } {PARA 2 "> " 1 "" {TEXT -1 27 "Error, (in f) invalid x: -3" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lasterror;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6$Q.invalid~x:~%16\"!\"$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lastexception;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6%%\"fGQ.invalid~x:~%16\"!\"$ "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "try\n f(-3):\n printf(\"%s\\n\",\"no error occurred\")\ncatch \"invalid x\":\n printf(\"%s\\n\",\"error occurred\")\ncatch :\n error\nend try;" } } {PARA 2 "> " 1 "" {TEXT -1 14 "error occurred" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " # Backward compatibility of ERROR:\nERROR(`invalid arguments`);" } } {PARA 2 "> " 1 "" {TEXT -1 24 "Error, invalid arguments" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lasterror;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%2invalid~argumentsG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lastexception;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6$\"\"!Q2invalid~arguments6\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ERROR(`invalid arguments`,2,x);" } } {PARA 2 "> " 1 "" {TEXT -1 30 "Error, invalid arguments, 2, x" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lasterror;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6%%2invalid~argumentsG\"\"#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lastexception;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6&\"\"!Q6invalid~arguments,~%06\"\"\"#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ERROR(2,x);" } } {PARA 2 "> " 1 "" {TEXT -1 11 "Error, 2, x" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lasterror;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6$\"\"#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lastexception;" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6&\"\"!Q#%06\"\"\"#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "StringTools:-FormatMessage( %[2..-1] );" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#Q%2,~x6\" "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "lasterror" 2 "lasterror" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Procedures" 2 "procedure" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "return" 2 "return" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "StringTools[FormatMessage]" 2 "StringTools[FormatMessage]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "try" 2 "try" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "WARNING" 2 "WARNING" "" }{TEXT -1 1 " " } } }} {PARA 11 "> " 1 "" {XPPMATH 20 "6&\"\"!Q6invalid~arguments,~%06\"\"\"#%\"xG "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ER{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 4 "Eval" } {TEXT 30 25 " - Evaluate an expression" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 10 " Eval(" } {TEXT 35 1 "a" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 1 "=" } {TEXT 35 1 "n" } {TEXT -1 2 "))" } } {PARA 0 "> " 0 "" {TEXT -1 10 " Eval(" } {TEXT 35 1 "a" } {TEXT -1 3 ", {" } {TEXT 35 2 "x1" } {TEXT -1 1 "=" } {TEXT 35 2 "n1" } {TEXT -1 1 "," } {TEXT 35 2 "x2" } {TEXT -1 1 "=" } {TEXT 35 2 "n2" } {TEXT -1 6 ",...})" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 16 "a - " } {TEXT -1 10 "expression" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 16 "x, x1, x2,... - " } {TEXT -1 5 "names" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 16 "n, n1, n2,... - " } {TEXT -1 17 "evaluat} } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "with(networks):\nG := complete(4):\naddvertex(0,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#\"\"! "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "connect(0,1,G,directed);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#%#e7G "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "arrivals(0,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "departures(0,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<#\"\"\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "neighbors(0,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<#\"\"\" "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "arrivals(1,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<&\"\"!\"\"#\"\"$\"\"% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "departures(1,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<%\"\"#\"\"$\"\"% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "neighbors(1,G);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#<&\"\"!\"\"#\"\"$\"\"% "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "arrivals(G);" } } {PARA/sjgfalsegoptruegygfzgfxgfgpropzeroconsidercontrastnoteinsidoccurdoalldefaultspecificomputatassociatusedvariablorderconstantcontainrespectiveitselfgeneratidealwhethequivalentfieldclosoversolutleastatbasidegretotalusingcommandstructurdatapolynomialidealviewrespectwithindeterminatoptionalpolynomiallistsetdetailindexisalgebraicalconsistalgebraicalsystemalgebraicgivendecidispropgroebntSregfopengarccotgarccotarcsaxiscutarcsincutsbranchfunctionadvisorunderstoodaroundlikeconstructinfinityginityginfrealrangegingrealranginfinitimgfregandgcomplexrangegzgcomplexrangimassumimplicitinstancsayyouconventmaplnotefoundfunctexprindexusingexpressintorangrealcomplexrelationrelatconvertڢ{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 38 "An Escape to the Host Operating System" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 2 "! " } {TEXT 35 7 "command" } {TEXT -1 1 " " } } {PARA 15 "> " 0 "" {TEXT -1 14 "The character " } {TEXT 35 1 "!" } {TEXT -1 96 " when it appears as the first character in a line is treated as an ``escape to host'' operator. " } } {PARA 15 "> " 0 "" {TEXT -1 182 "This allows the execution of host system commands from within a Maple session. All characters on the line following the escape character are treated as a command to the host system. " } } {PARA 15 "> " 0 "" {TEXT -1 53 "This does not return any value to the Maple session. " } } {PARA 15 "> " 0 "" {TEXT -1 79 "Note: There may be implementation restrictions on some host operating systems. " } } } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "backslash" 2 "backslash" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "system" 2 "system" "" }{TEXT -1 2 " " } } }} " 0 "" {TEXT -1 38 "An Escape to the Host Operating System" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 2 "! " } {TEXT 35 7 "command" } {TEXT -1 1 " " } } {PARA 15 "> " 0 "" {TEXT -1 14 "The character " } {TEXT 35 1 "!" } {TEXT -1 96 " when it appears as the first character in a line is treated as an ``escape to host'' operator. " } } {PARA 15 "> " 0 "" {TEXT -1 182 "This allows the execution of host system commands from within a Maple session. All characters on the line following the escape character are treated as a command to the host system. " } } {PARA 15 "> " 0 "" {TEXT -1 53 "This does not return any value to the Maple session. " } } {PARA 15 "> " 0 "" {TEXT -1 79 "Note: There may be implementation restrictions on some host operating systems. " } } } {SECT 0 {PARA 4 "> " 0 "are sorted into ``address'' order and coefficients are automatically extracted, " } {TEXT 35 14 "(d(x) &^ d(t))" } {TEXT -1 36 " is recognized as being the same as " } {TEXT 35 16 "- (d(t) &^ d(x))" } {TEXT -1 2 ". " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(liesymm): setup(x,y,z):" } } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "wcollect( a*(d(x) &^ d(y)) + b*(d(x) &^ d(y)));" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"%\"bGF&F&-%#&^G6$-%\"dG6#%\"xG-F,6#%\"yGF& "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "wcollect( a*(d(x) &^ d(y)) + b*(d(y) &^ d(x)));" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"%\"bG!\"\"F&-%#&^G6$-%\"dG6#%\"xG-F-6#%\"yGF& "}}} {PARA 0 "> " 0 "" {TEXT -1 1 " " } } } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "with" 2 "with" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "liesymm" 2 "liesymm" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "liesymm[setup]" 2 "liesymm[setup],)Z#zgrootofgrgkmwdgcfkotghtqgduringprintinformatsomepositsettgivenmessagwarnworkdoesindexdividcomputatsymbolicusingoccurramificatwildnotryadicspringcohenhenritheorcomputationalcoursroundtwobordeauxnombrdestheoridejournalletardordermaximalimplementdefaultusedalgorithmfourroundzassenhaufordroundfourmethodinsteadthesatlocalcaseprimvaluformsupportfunctvariabltermexpressdiffmaybutalsowillinputwithmonicannihilatcallelemintegerringdenotmodulelementlistoutputisomorphismuprootgeneratindependappearcoefficientnotatradicalrationalmustrepresentrootofconsideratunderspecifwaysseveralcomputproceduroptionoverpolynomialirreduciblunivariatnumbersetoptfieldnumbalgebraicbasebasiintegralnumtheor ". " } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 1 "n" } {TEXT -1 20 "th Euler polynomial " } {TEXT 35 7 "E(n, x)" } {TEXT -1 52 " is defined by the exponential generating function: " } } {PARA 0 "> " 0 "" {TEXT 23 68 " 2*exp(x*t)/(exp(t)+1) = sum(E(n,x)/n!*t^n, n = 0..infinity)" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 1 "n" } {TEXT -1 16 "th Euler number " } {TEXT 35 4 "E(n)" } {TEXT -1 52 " is defined by the exponential generating function: " } } {PARA 0 "> " 0 "" {TEXT 23 63 " 2/(exp(t)+exp(-t)) = sum(E(n)/n!*t^n, n = 0..infinity)" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 1 "n" } {TEXT -1 20 "th Euler polynomial " } {TEXT 35 7 "E(n, x)" } {TEXT -1 9 " and the " } {TEXT 35 1 "n" } {TEXT -1 16 "th Euler number " } {TEXT 35 4 "E(n)" } {TEXT -1 40 " are related by the following equation: " } } {PARA 0 "> " 0 "" {TEXT 23 28 " E(n) = 2^n*E(n,1/2)" } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "euler(6);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#!#h "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "euler(4,x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,(*$%\"xG\"\"%\"\"\"*$F%\"\"$!\"#F%F' "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "euler(4,1/2);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6##\"\"&\"#; "}}} } {SECT 0 {PARA 4 "> " 0 "seealso" {TEXT -1 9 "See Also " } } {PARA 0 "> " 0 "" {HYPERLNK 17 "bernoulli" 2 "bernoulli" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "inifcns" 2 "inifcns" "" }{TEXT -1 1 " " } } }} 23 63 " 2/(exp(t)+exp(-t)) = sum(E(n)/n!*t^n, n = 0..infinity)" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 1 "n" } {TEXT -1 20 "th Euler polynomial " } {TEXT 35 7 "E(n, x)" } {TEXT -1 9 " and the " } {TEXT 35 1 "n" } {TEXT -1 16 "th Euler number " } {TEXT 35 4 "E(n)" } {TEXT -1 40 " are related by the following equation: " } } {PARA 0 "> " 0 "" {TEXT 23 28 " E(n) = 2^n*E(n,1/2)" } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eu xfbefargumusefulnesillustratexampldececonstructspecialbutfollowzgfrootofgfailgygfagfafactorfalsegxgfagrootofaliatruegygxgfggeometrmethodeffectmegaragotfrancoijeanprobabilisticreferencfactorcanlikeverycontrollfailurprobabilitalthoughchosengreatfiveoddtenfirstgiventhroughrunsmoduloreductsoughtcriterrationaltrivialnonrationalcaseotherwisfailreduciblfalsdetecttruereturnreducibilitconditsufficilookdegreonlyirreduciblabsoluteunivariatnotatcoefficienthavemustnumbercomplexwithconjunctusedcoefficiclosurtestplaceholdintegerprimlistsetoptionalmultivariatdomainuniqufieldnumbalgebraicoverfactorizatstandoutformpolynomialirreducibilitabsolutinertfunctindexairreducevala {VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 8 "AIrreduc" } {TEXT 30 41 " - inert absolute irreducibility function" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 14 " AIrreduc(" } {TEXT 35 1 "P" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 14 " AIrreduc(" } {TEXT 35 1 "P" } {TEXT -1 2 ", " } {TEXT 35 1 "S" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 4 "P - " } {TEXT -1 24 "multivariate polynomial " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 4 "S - " } {TEXT -1 40 "(optional) set or list of prime integers" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 4 "The " } {TEXT 35 8 "AIrreduc" } {TEXT -1 85 " function is a placeholder for testing the absolute irreducibility of the polynomial"B {VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 15 "evalhf/constant" } {TEXT 30 28 " - constants known to evalhf" } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 37 "The following constants are known to " } {TEXT 35 6 "evalhf" } {TEXT -1 2 ": " } } {PARA 17 "> " 0 "wmitable" {TEXT -1 220 "Pi gamma Catalan true \nfalse FLT_RADIX DBL_MANT_DIG DBL_DIG \nDBL_EPSILON DBL_MIN_EXP DBL_MIN DBL_MIN_10_EXP \nDBL_MAX_EXP DBL_MAX DBL_MAX_10_EXP LNMAXFLOAT " } } {PARA 0 "> " 0 "" {TEXT -1 1 " " } } {PARA 15 "> " 0 "" {TEXT 35 14 "evalhf(Digits)" } {TEXT -1 124 " returns the approximate number of digits of the hardware floating point system in use, regardless of the actual setting of " } {TEXT 35 7 "Digits." } {TEXT -1 97 " This constant takes into account the precisi b{VERSION 4 0 "HELP" "4.0"} {USTYLETAB {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 10 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "> " 0 "" {TEXT -1 8 "eulermac" } {TEXT 30 28 " - Euler-Maclaurin summation" } } {PARA 4 "> " 0 "usage" {TEXT -1 16 "Calling Sequence" } } {PARA 0 "> " 0 "" {TEXT -1 14 " eulermac(" } {TEXT 35 4 "expr" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 18 ") or eulermac(" } {TEXT 35 4 "expr" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 3 " = " } {TEXT 35 1 "a" } {TEXT -1 2 ".." } {TEXT 35 1 "b" } {TEXT -1 1 ")" } } {PARA 0 "> " 0 "" {TEXT -1 14 " eulermac(" } {TEXT 35 4 "expr" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 2 ", " } {TEXT 35 1 "n" } {TEXT -1 16 ") or eulermac(" } {TEXT 35 4 "expr" } {TEXT -1 2 ", " } {TEXT 35 1 "x" } {TEXT -1 3 " = " } {TEXT 35 1 "a" } {TEXT -1 2 ".." } {TEXT 35 1 "b" } {TEXT -1 2 ", " } {TEXT 35 1 "n" } {TEXT -1 1 ")" } } {PARA 4 "> " 0 "" {TEXT -1 10 "Parameters" } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "expr - " } {TEXT -1 14 "expression in " } {TEXT 35 1 "x" } {TEXT -1 1 " " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "x - " } {TEXT -1 21 "independent variable " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "a,b - " } {TEXT -1 61 "interval over which the approximation to the sum is computed " } } {PARA 0 "> " 0 "" {TEXT -1 5 " " } {TEXT 23 7 "n - " } {TEXT -1 41 "(optional) integer (degree of summation) " } } {SECT 0 {PARA 4 "> " 0 "info" {TEXT -1 11 "Description" } } {PARA 15 "> " 0 "" {TEXT -1 10 "The forms " } {TEXT 35 17 "eulermac(expr, x)" } {TEXT -1 5 " and " } {TEXT 35 20 "eulermac(expr, x, n)" } {TEXT -1 38 " compute asymptotic approximations to " } {TEXT 35 12 "sum(expr, x)" } {TEXT -1 5 ". If " } {TEXT 35 24 "F(x) = eulermac(f(x), x)" } {TEXT -1 7 ", then " } {TEXT 35 15 "F(x + 1) - F(x)" } {TEXT -1 33 " is asymptotically equivalent to " } {TEXT 35 4 "f(x)" } {TEXT -1 49 ". The order of the approximation is specified by " } {TEXT 35 1 "n" } {TEXT -1 18 ", and defaults to " } {TEXT 35 9 "Order - 1" } {TEXT -1 2 ". " } } {PARA 15 "> " 0 "" {TEXT -1 10 "The forms " } {TEXT 35 22 "eulermac(expr, x=a..b)" } {TEXT -1 5 " and " } {TEXT 35 25 "eulermac(expr, x=a..b, n)" } {TEXT -1 9 " compute " } {TEXT 35 1 "n" } {TEXT -1 10 "th degree " } {HYPERLNK 45 "Euler-Maclaurin summation formulas" 5 "Definition/Eulersummationformula" "" }{TEXT -1 5 " for " } {TEXT 35 4 "expr" } {TEXT -1 7 " (thus " } {TEXT 35 1 "n" } {TEXT -1 39 " terms of the expansion are given). If " } {TEXT 35 1 "n" } {TEXT -1 39 " is not specified, it is assumed to be " } {TEXT 35 9 "Order - 1" } {TEXT -1 2 ". " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "\neulermac( 1/x, x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,.-%#lnG6#%\"xG\"\"\"*$F'!\"\"#F*\"\"#*$F'!\"##F*\"#7*$F'!\"%#F(\"$?\"* $F'!\"'#F*\"$_#-%\"OG6#*$F'!\")F( "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eulermac( 1/k, k=1..x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,0-%$intG6$*$%\"kG!\"\"/F(;\"\"\"%\"xGF,%&gammaGF,*$F-F)#F,\"\"#*$F -!\"##F)\"#7*$F-!\"%#F,\"$?\"*$F-!\"'#F)\"$_#-%\"OG6#*$F-!\")F, "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eulermac( 1/x^2, x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,.*$%\"xG!\"\"F&*$F%!\"##F&\"\"#*$F%!\"$#F&\"\"'*$F%!\"&#\"\"\"\"#I*$F%! \"(#F&\"#U-%\"OG6#*$F%!\"*F2 "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eulermac(1/k^2, k=1..x, 4);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,0-%$intG6$*$%\"kG!\"#/F(;\"\"\"%\"xGF,*$%#PiG\"\"##F,\"\"'!\"\"F,*$F- F)#F,F0*$F-!\"$#F3F2*$F-!\"&#F,\"#I-%\"OG6#*$F-!\"(F, "}}} {PARA 0 "> " 0 "" {TEXT -1 1 " " } } }} T -1 2 ". " } } } {SECT 0 {PARA 4 "> " 0 "examples" {TEXT -1 8 "Examples" } } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "\neulermac( 1/x, x);" } } {PARA 11 "> " 1 "" {XPPMATH 20 "6#,.-%#lnG6#%\"xG\"\"\"*$F'!\"\"#F*\"\"#*$F'!\"##F*\"#7*$F'!\"%#F(\"$?\"* $F'!\"'#F*\"$_#-%\"OG6#*$F'!\")F( "}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eulermac( 1/k, k=1..x);" } } {PARA 11 "> " 1 "" {XPPMAT~pigsingpisinirreduciblallpretendsubstitutdetectpreproczgfsomegeneratreduciblzgrootofgygfalphagfygalphagsqrtalphaaliaxgfxgaccordingchanganothsessvaryobjectsincproceedcomputatbeforrecursiveargumenttheirusecheckindependencskipcanbasisatisfineedconditreturnerrorotherwisquantitsumdistributalwaybecausaccuratperfectthereforautomaticalmaplcanonicalactualelementpresfollowexampleventermnontrivialcoefficientrationalizateachproductpartialformalwellprimrelativetheymoreovunitabovgeneralvariabllccoefficileadsayabusivenoteoutbelowcaseexceptionalexceptcontinteghavebothnodenominatoralgebraemptdegrelesspositmustoutputwordpolynomialminimalmoduloreducpowerviewintegerringalsoshallwesubcontaindoeschosenorderelemconsiderexprdenotpureothersuchletindeterm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d: bni64.h,v 1.1.1.1 1999/08/08 19:38:15 heller Exp $ */ #ifndef Included_bni64_h #define Included_bni64_h #include "bni.h" #include "pgpMemoryMgr.h" #ifndef BNWORD64 #error 64-bit bignum library requires a 64-bit data type #endif PGP_BEGIN_C_DECLARATIONS #ifndef bniCopy_64 void bniCopy_64(BNWORD64 *dest, BNWORD64 const *src, unsigned len); #endif #ifndef bniZero_64 void bniZero_64(BNWORD64 *num, unsigned len); #endif #ifndef bniNeg_64 void bniNeg_64(BNWORD64 *num, unsigned len); #endif #ifndef bniAdd1_64 BNWORD64 bniAdd1_64(BNWORD64 *num, unsigned len, BNWORD64 carry); #endif #ifndef bniSub1_64 BNWORD64 bniSub1_64(BNWORD64 *num, unsigned len, BNWORD64 borrow); #endif #ifndef bniAddN_64 BNWORD64 bniAddN_64(BNWORD64 *num1, BNWORD64 const *num2, unsigned len); #endif #ifndef bniSubN_64 BNWORD64 bniSubN_64(BNWORD64 *num1, BNWORD64 const *num2, unsigned len); #endif #ifndef bniCmp_64 int bniCmp_64(BNWORD64 const *num1, BNWORD64 const *num2, unsigned len); #endif #ifndef bniMulN1_64 void bniMulN1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k); #endif #ifndef bniMulAdd1_64 BNWORD64 bniMulAdd1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k); #endif #ifndef bniMulSub1_64 BNWORD64 bniMulSub1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k); #endif #ifndef bniLshift_64 BNWORD64 bniLshift_64(BNWORD64 *num, unsigned len, unsigned shift); #endif #ifndef bniDouble_64 BNWORD64 bniDouble_64(BNWORD64 *num, unsigned len); #endif #ifndef bniRshift_64 BNWORD64 bniRshift_64(BNWORD64 *num, unsigned len, unsigned shift); #endif #ifndef bniMul_64 void bniMul_64(BNWORD64 *prod, BNWORD64 const *num1, unsigned len1, BNWORD64 const *num2, unsigned len2); #endif #ifndef bniSquare_64 void bniSquare_64(BNWORD64 *prod, BNWORD64 const *num, unsigned len); #endif #ifndef bniNorm_64 unsigned bniNorm_64(BNWORD64 const *num, unsigned len); #endif #ifndef bniBits_64 unsigned bniBits_64(BNWORD64 const *num, unsigned len); #endif #ifndef bniExtractBigBytes_64 void bniExtractBigBytes_64(BNWORD64 const *bn, unsigned char *buf, unsigned lsbyte, unsigned buflen); #endif #ifndef bniInsertBigytes_64 void bniInsertBigBytes_64(BNWORD64 *n, unsigned char const *buf, unsigned lsbyte, unsigned buflen); #endif #ifndef bniExtractLittleBytes_64 void bniExtractLittleBytes_64(BNWORD64 const *bn, unsigned char *buf, unsigned lsbyte, unsigned buflen); #endif #ifndef bniInsertLittleBytes_64 void bniInsertLittleBytes_64(BNWORD64 *n, unsigned char const *buf, unsigned lsbyte, unsigned buflen); #endif #ifndef bniDiv21_64 BNWORD64 bniDiv21_64(BNWORD64 *q, BNWORD64 nh, BNWORD64 nl, BNWORD64 d); #endif #ifndef bniDiv1_64 BNWORD64 bniDiv1_64(BNWORD64 *q, BNWORD64 *rem, BNWORD64 const *n, unsigned len, BNWORD64 d); #endif #ifndef bniModQ_64 unsigned bniModQ_64(BNWORD64 const *n, unsigned len, unsigned d); #endif #ifndef bniDiv_64 BNWORD64 bniDiv_64(BNWORD64 *q, BNWORD64 *n, unsigned nlen, BNWORD64 *d, unsigned dlen); #endif #ifndef bniMontInv1_64 BNWORD64 bniMontInv1_64(BNWORD64 const x); #endif #ifndef bniMontReduce_64 void bniMontReduce_64(BNWORD64 *n, BNWORD64 const *mod, unsigned const mlen, BNWORD64 inv); #endif #ifndef bniToMont_64 void bniToMont_64(BNWORD64 *n, unsigned nlen, BNWORD64 *mod, unsigned mlen); #endif #ifndef bniFromMont_64 void bniFromMont_64(BNWORD64 *n, BNWORD64 *mod, unsigned len); #endif #ifndef bniExpMod_64 int bniExpMod_64( PGPMemoryMgrRef mgr, PGPBoolean secure, BNWORD64 *result, BNWORD64 const *n, unsigned nlen, BNWORD64 const *exp, unsigned elen, BNWORD64 *mod, unsigned mlen); #endif #ifndef bniDoubleExpMod_64 int bniDoubleExpMod_64( PGPMemoryMgrRef mgr, PGPBoolean secure, BNWORD64 *result, BNWORD64 const *n1, unsigned n1len, BNWORD64 const *e1, unsigned e1len,