A flexible conversion facility in the mathematical language is as important as a dictionary in spoken languages. This page describes such a tool, implemented as a net of conversion routines permitting the expression of any mathematical function in terms of another one, whenever that is possible as a finite sum of terms with rational coefficients. When converting to special functions these coefficients may also involve elementary functions. When the parameters of the functions being converted depend on symbols in a rational manner, any assumptions on these symbols made by the user are taken into account at the time of performing the conversions (see assuming).

The first argument given to convert, expr, is the object being converted. expr can be any Maple mathematical object, such as an expression, equation, a list or set of them.

The second argument, SF, is the name of a mathematical function or that of a function class. The conversion routines attempt to express expr in terms of SF.

The rewriting of expr in terms of SF is performed in two sequential steps, by attempting first a Global and then a Local conversion. A Global conversion is one where the expr to be converted is treated as a whole. Global conversions are implemented for only few target SF functions and typically result in simpler output expressions. A Local conversion is one where only the mathematical functions found in expr are rewritten in terms of SF. Local conversions are implemented for most Maple mathematical functions (see table below).

Two given functions - say, F and G - could in principle be expressed in terms of each other if they belong to the same function class. That condition, however, is in many cases not sufficient: F may be more general than G, or simply the conversion may only be possible for some particular values of the function parameters.

Some mathematical functions automatically evaluate to other functions. So, to perform the conversion to one of these functions, say F, it is sometimes necessary to delay the evaluation of F; see the Examples section.

The functions to which you can convert - the second argument could be one of these - are:

When a function class name is given as second argument the routines attempt to express expr using any of the functions of that class. The function classes understood by convert are:

* (note spaces between words are filled with _ )

Despite the large number of function classes, most of the functions of mathematical physics belong to one of the three hypergeometric classes: 2F1, 1F1, and 0F1 where these three classes also include as particular cases all the elementary functions (trig, hyperbolic trig, their arcs, exp, and ln).

That gives a superficial but straightforward and usually sufficient idea of how the mathematical functions are / could be related between themselves. This division into three sets also tells about the possible elementary form of the special functions of each of the classes 2F1, 1F1, and 0F1: since functions of each class can be transformed between themselves, and each of these three classes contain both special and elementary functions, the elementary form of the special functions of each class can only be obtained using the elementary functions belonging to the same class. For instance, 0F1 includes all the Bessel related and trigonometric functions, from where, for particular values of the parameters entering the Bessel functions, they can be expressed as sin or cos. In the same line, the 2F1 functions include the Legendre and arctrig ones, from where, for particular values of the parameters in the Legendre functions they admit an arctrig form.

$\mathrm{WhittakerM}\left(0\,\frac{1}{2}\,z\right)$

$\mathrm{WhittakerM}\left(0\,\frac{1}{2}\,z\right)$

$=\mathrm{convert}\left(\,\exp \right)$

$\mathrm{WhittakerM}\left(0\,\frac{1}{2}\,z\right)={\ⅇ}^{\frac{z}{2}}-\frac{1}{{\ⅇ}^{\frac{z}{2}}}$

$\mathrm{convert}\left(\,\mathrm{MeijerG}\right)$

$\frac{z\mathrm{MeijerG}\left(\left[\left[0\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[\mathrm{-1}\right]\right]\,-z\right)}{{\ⅇ}^{\frac{z}{2}}}={\ⅇ}^{\frac{z}{2}}-\frac{1}{{\ⅇ}^{\frac{z}{2}}}$

$\mathrm{map}\left(\mathrm{numer}\,\right)$

$z\mathrm{MeijerG}\left(\left[\left[0\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[\mathrm{-1}\right]\right]\,-z\right)={\left({\ⅇ}^{\frac{z}{2}}\right)}^2-1$

$\mathrm{convert}\left(\,\mathrm{hypergeom}\,\mathrm{include}=\exp \right)$

$z\mathrm{hypergeom}\left(\left[1\right]\,\left[2\right]\,z\right)={\mathrm{hypergeom}\left(\left[\right]\,\left[\right]\,\frac{z}{2}\right)}^2-1$

$\mathrm{simplify}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right)\right)$

$0$

$\mathrm{JacobiP}\left(\frac{1}{2}\,0\,-1\,z\right)$

$\mathrm{JacobiP}\left(\frac{1}{2}\,0\,\mathrm{-1}\,z\right)$

$\mathrm{convert}\left(\,\mathrm{EllipticE}\right)$

$\frac{2\mathrm{EllipticE}\left(\frac{\sqrt{-2z+2}}{2}\right)}{\mathrm{\pi }}$

$\mathrm{convert}\left(\,\mathrm{LegendreP}\right)$

$\frac{\left(1+z\right)\left(\mathrm{LegendreP}\left(-\frac{1}{2}\,z\right)+\mathrm{LegendreP}\left(\frac{1}{2}\,\frac{3-z}{1+z}\right)\sqrt{2}\sqrt{\frac{1}{1+z}}\right)}{4}$

$\mathrm{convert}\left(\,\mathrm{GegenbauerC}\right)$

$\frac{\left(1+z\right)\left(\mathrm{GegenbauerC}\left(-\frac{1}{2}\,\frac{1}{2}\,z\right)+\mathrm{GegenbauerC}\left(\frac{1}{2}\,\frac{1}{2}\,\frac{3-z}{1+z}\right)\sqrt{2}\sqrt{\frac{1}{1+z}}\right)}{4}$

$\mathrm{convert}\left(\,\mathrm{hypergeom}\right)$

$\frac{\left(1+z\right)\left(\mathrm{hypergeom}\left(\left[\frac{1}{2}\,\frac{1}{2}\right]\,\left[1\right]\,\frac{1}{2}-\frac{z}{2}\right)+\mathrm{hypergeom}\left(\left[-\frac{1}{2}\,\frac{3}{2}\right]\,\left[1\right]\,\frac{z-1}{1+z}\right)\sqrt{2}\sqrt{\frac{1}{1+z}}\right)}{4}$

$\frac{\frac{1}{2}\mathrm{LegendreP}\left(-\frac{1}{2}\,-\frac{1}{2}\,-1+4z-2z^2\right){\left(4z-2z^2\right)}^{\frac{1}{4}}}{{\left(-2{\left(-1+z\right)}^2\right)}^{\frac{1}{4}}}\left(-1+z\right)-\frac{\arcsin \left(-1+z\right)}{{\mathrm{\pi }}^{\frac{1}{2}}}$

$\frac{\mathrm{LegendreP}\left(-\frac{1}{2}\,-\frac{1}{2}\,-2z^2+4z-1\right){\left(-2z^2+4z\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(z-1\right)}{2{\left(-2{\left(z-1\right)}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}-\frac{\arcsin \left(z-1\right)}{\sqrt{\mathrm{\pi }}}$

$\mathrm{convert}\left(\,\mathrm{arctrig}\right)$

$0$

$\mathrm{convert}\left(\,\mathrm{hypergeom}\,\mathrm{include}=\mathrm{elementary}\right)$

$0$

$\mathrm{erf}\left(z\right)$

$\mathrm{erf}\left(z\right)$

$\mathrm{convert}\left(\,\mathrm{HermiteH}\right)$

$-\frac{2\mathrm{HermiteH}\left(\mathrm{-1}\,z\right)}{\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}}+1$

$\mathrm{convert}\left(\,\mathrm{KummerU}\right)$

$-\frac{2\left(\frac{z\mathrm{KummerU}\left(1\,\frac{3}{2}\,z^2\right)}{2}-\frac{\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}\left(z-\sqrt{z^2}\right)}{2\sqrt{z^2}}\right)}{\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}}+1$

$\mathrm{evala}\left(\mathrm{normal}\left(\mathrm{convert}\left(\,\mathrm{WhittakerW}\right)\right)\right)$

$\frac{\left({\left(z^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}-\mathrm{WhittakerW}\left(-\frac{1}{4}\,\frac{1}{4}\,z^2\right){\ⅇ}^{\frac{z^2}{2}}\right){\left(z^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\ⅇ}^{z^2}z\sqrt{\mathrm{\pi }}}$

$\mathrm{convert}\left(\,\mathrm{hypergeom}\right)$

$\frac{\left({\left(z^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}-{\left(z^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{\mathrm{\pi }}\mathrm{hypergeom}\left(\left[\right]\,\left[\right]\,z^2\right)+2{\left(z^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{hypergeom}\left(\left[1\right]\,\left[\frac{3}{2}\right]\,z^2\right)\right){\left(z^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\ⅇ}^{z^2}z\sqrt{\mathrm{\pi }}}$

$\mathrm{simplify}\left(\right)$

$\mathrm{erf}\left(z\right)$

$\mathrm{Ai}≔\mathrm{AiryAi}\left(z\right)$

$\mathrm{Ai}≔\mathrm{AiryAi}\left(z\right)$

$\mathrm{Ai}=\mathrm{convert}\left(\mathrm{Ai}\,\mathrm{BesselJ}\right)$

$\mathrm{AiryAi}\left(z\right)=\frac{{\left({\left(-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{BesselJ}\left(-\frac{1}{3}\,\frac{2{\left(-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)}{3}-\frac{z\mathrm{BesselJ}\left(\frac{1}{3}\,\frac{2{\left(-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)}{3{\left({\left(-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

$\mathrm{Ai}=\mathrm{convert}\left(\mathrm{Ai}\&comma;\mathrm{BesselJ}\right)\mathbf{assuming}\mathrm{\Re }\left(z\right)<0$

$\mathrm{AiryAi}\left(z\right)=\frac{\sqrt{-z}\left(\mathrm{BesselJ}\left(\frac{1}{3}\&comma;\frac{2{\left(-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)+\mathrm{BesselJ}\left(-\frac{1}{3}\&comma;\frac{2{\left(-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)\right)}{3}$

$\mathrm{Ai}=\mathrm{convert}\left(\mathrm{Ai}\&comma;\mathrm{BesselK}\right)\mathbf{assuming}0<\mathrm{\Re }\left(z\right)$

$\mathrm{AiryAi}\left(z\right)=\frac{\sqrt{3}\sqrt{z}\mathrm{BesselK}\left(\frac{1}{3}\&comma;\frac{2z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)}{3\mathrm{\pi }}$

$\mathrm{Ai}=\mathrm{convert}\left(\mathrm{Ai}\&comma;\mathrm{KummerU}\right)\mathbf{assuming}0<\mathrm{\Re }\left(z\right)$

$\mathrm{AiryAi}\left(z\right)=\frac{3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\sqrt{z}{4}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{KummerU}\left(\frac{5}{6}\&comma;\frac{5}{3}\&comma;\frac{4z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)}{3\sqrt{\mathrm{\pi }}{\&ExponentialE;}^{\frac{2z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}}}$

$\mathrm{Ai}=\mathrm{convert}\left(\mathrm{Ai}\&comma;\mathrm{WhittakerW}\right)\mathbf{assuming}0<\mathrm{\Re }\left(z\right)$

$\mathrm{AiryAi}\left(z\right)=\frac{\sqrt{z}\mathrm{WhittakerW}\left(0\&comma;\frac{1}{3}\&comma;\frac{4z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{3}\right)}{2\sqrt{\mathrm{\pi }}\sqrt{z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

$\mathrm{HermiteH}\left(a\&comma;z\right)$

$\mathrm{HermiteH}\left(a\&comma;z\right)$

$=\mathrm{convert}\left(\&comma;\mathrm{HermiteH}\&comma;''raise\; a''\right)$

$\mathrm{HermiteH}\left(a\&comma;z\right)=\frac{z\mathrm{HermiteH}\left(a+1\&comma;z\right)-\frac{\mathrm{HermiteH}\left(a+2\&comma;z\right)}{2}}{a+1}$

$\mathrm{ee}≔\frac{1}{8}\mathrm{KummerU}\left(\frac{3}{2}\&comma;\frac{1}{2}\&comma;z^2\right)-\frac{1}{8}\mathrm{KummerU}\left(1\&comma;\frac{3}{2}\&comma;z^2\right)\mathrm{sqrt}\left(z^2\right)-\frac{1}{4}\mathrm{KummerU}\left(1\&comma;\frac{3}{2}\&comma;z^2\right)z^2\mathrm{sqrt}\left(z^2\right)+\frac{1}{4}\mathrm{sqrt}\left(z^2\right)$

$\mathrm{ee}≔\frac{\mathrm{KummerU}\left(\frac{3}{2}\&comma;\frac{1}{2}\&comma;z^2\right)}{8}-\frac{\mathrm{KummerU}\left(1\&comma;\frac{3}{2}\&comma;z^2\right)\sqrt{z^2}}{8}-\frac{\mathrm{KummerU}\left(1\&comma;\frac{3}{2}\&comma;z^2\right)z^2\sqrt{z^2}}{4}+\frac{\sqrt{z^2}}{4}$

$\mathrm{convert}\left(\mathrm{ee}\&comma;\mathrm{KummerU}\&comma;''normalize\; a''\&comma;''normalize\; b''\&comma;''mix\; a\; and\; b''\right)$

$\frac{\sqrt{z^2}\mathrm{KummerU}\left(2\&comma;\frac{3}{2}\&comma;z^2\right)}{8}-\frac{\mathrm{KummerU}\left(\frac{1}{2}\&comma;\frac{1}{2}\&comma;z^2\right)}{8}-\frac{\mathrm{KummerU}\left(\frac{1}{2}\&comma;\frac{1}{2}\&comma;z^2\right)z^2}{4}+\frac{\sqrt{z^2}}{4}$

$\mathrm{factor}\left(\mathrm{convert}\left(\&comma;\mathrm{KummerU}\&comma;''normalize\; a''\&comma;''normalize\; b''\&comma;''mix\; a\; and\; b''\right)\right)$

$-\frac{\left(2z^2+1\right)\left(\mathrm{KummerU}\left(1\&comma;\frac{3}{2}\&comma;z^2\right)\sqrt{z^2}-\mathrm{KummerU}\left(\frac{1}{2}\&comma;\frac{1}{2}\&comma;z^2\right)\right)}{8}$

$\mathrm{convert}\left(\&comma;\mathrm{KummerU}\&comma;''normalize\; a''\&comma;''normalize\; b''\&comma;''mix\; a\; and\; b''\right)$

$0$

$\mathrm{H1}≔\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;z\right)$

$\mathrm{H1}≔\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;z\right)$

$\mathrm{ee}≔\mathrm{H1}=\mathrm{convert}\left(\mathrm{H1}\&comma;\mathrm{hypergeom}\&comma;''raise\; a''\right)$

$\mathrm{ee}≔\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;z\right)=\frac{\left(\left(-z+2\right)a+\left(b-1\right)z-c+2\right)\mathrm{hypergeom}\left(\left[b\&comma;a+1\right]\&comma;\left[c\right]\&comma;z\right)+\mathrm{hypergeom}\left(\left[b\&comma;a+2\right]\&comma;\left[c\right]\&comma;z\right)\left(z-1\right)\left(a+1\right)}{-c+a+1}$

$\mathrm{simplify}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\mathrm{ee}\right)\right)$

$0$

$\mathrm{lhs}\left(\mathrm{ee}\right)=\mathrm{collect}\left(\mathrm{convert}\left(\mathrm{rhs}\left(\mathrm{ee}\right)\&comma;\mathrm{hypergeom}\&comma;''lower\; c''\right)\&comma;\mathrm{hypergeom}\&comma;\mathrm{normal}\right)$

$\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;z\right)=\frac{\left(za-bz-2a+c+z-2\right)\left(c-1\right)\left(c-2\right)\left(z-1\right)\mathrm{hypergeom}\left(\left[b\&comma;a+1\right]\&comma;\left[c-2\right]\&comma;z\right)}{\left(-c+1+b\right)\left(a+2-c\right)z\left(-c+a+1\right)}+\frac{\left(za-bz-2a+c+z-2\right)\left(c-1\right)\left(za+bz-2zc+c+4z-2\right)\mathrm{hypergeom}\left(\left[b\&comma;a+1\right]\&comma;\left[c-1\right]\&comma;z\right)}{\left(-c+1+b\right)\left(a+2-c\right)z\left(-c+a+1\right)}-\frac{\left(c-1\right)\left(c-2\right){\left(z-1\right)}^2\left(a+1\right)\mathrm{hypergeom}\left(\left[b\&comma;a+2\right]\&comma;\left[c-2\right]\&comma;z\right)}{\left(-c+1+b\right)\left(-c+3+a\right)z\left(-c+a+1\right)}-\frac{\left(c-1\right)\left(za+bz-2zc+c+5z-2\right)\left(z-1\right)\left(a+1\right)\mathrm{hypergeom}\left(\left[b\&comma;a+2\right]\&comma;\left[c-1\right]\&comma;z\right)}{\left(-c+1+b\right)\left(-c+3+a\right)z\left(-c+a+1\right)}$

$\mathrm{simplify}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right)\right)$

$0$

$G≔\cos \left(z\sin \left(x\right)\right)$

$G≔\cos \left(z\sin \left(x\right)\right)$

$\mathrm{convert}\left(G\&comma;\exp \right)$

$\frac{{\&ExponentialE;}^{\frac{z\left({\&ExponentialE;}^{\mathrm{I}x}-{\&ExponentialE;}^{\mathrm{-I}x}\right)}{2}}}{2}+\frac{{\&ExponentialE;}^{-\frac{z\left({\&ExponentialE;}^{\mathrm{I}x}-{\&ExponentialE;}^{\mathrm{-I}x}\right)}{2}}}{2}$

$\mathrm{convert}\left(G\&comma;\exp \&comma;\mathrm{exclude}=\sin \right)$

$\frac{{\&ExponentialE;}^{\mathrm{I}z\sin \left(x\right)}}{2}+\frac{{\&ExponentialE;}^{\mathrm{-I}z\sin \left(x\right)}}{2}$

$\mathrm{convert}\left(G\&comma;\exp \&comma;\mathrm{only}=\sin \right)$

$\cosh \left(\frac{z\left({\&ExponentialE;}^{\mathrm{I}x}-{\&ExponentialE;}^{\mathrm{-I}x}\right)}{2}\right)$

$\mathrm{convert}\left(G\&comma;\exp \&comma;\left\{z\right\}\right)$

$\frac{{\&ExponentialE;}^{\mathrm{I}z\sin \left(x\right)}}{2}+\frac{{\&ExponentialE;}^{\mathrm{-I}z\sin \left(x\right)}}{2}$

$\mathrm{convert}\left(\sin \left(x\right)\&comma;\sinh \right)$

$\mathrm{-I}\sinh \left(\mathrm{I}x\right)$

$\mathrm{eval}\left(\right)$

$\sin \left(x\right)$

Cheb-Terrab, E.S. "The function advisor project: A Computer Algebra Handbook of Special Functions", Proceedings of the Maple Summer Workshop, University of Waterloo, Ontario, Canada, 2002.