Mathematical Functions - Maple Help

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 Mathematical Functions

Maple provides a state-of-the-art environment for algebraic and numeric computations with mathematical functions. The requirements concerning mathematical functions, however, are not just computational: typically, you also need information on identities, alternative definitions and mathematical properties in general. For these purposes Maple provides the MathematicalFunctions package and the FunctionAdvisor command, whose main goals are to provide tools for advanced computations with mathematical functions, and to make the information that the Maple system can provide more complete at each release, providing access to each piece of information through a simple interface.

 • For Maple 2015, an important amount of mathematical formulas were added to the database of the FunctionAdvisor   command.

Examples

 • The complex components:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{argument}\right)$
 $\left[{\mathrm{arg}}{}\left({z}\right){=}{-I}{}{\mathrm{ln}}{}\left(\frac{{z}}{\left|{z}\right|}\right){,}{\mathrm{arg}}{}\left({z}\right){=}{-I}{}{\mathrm{ln}}{}\left({\mathrm{signum}}{}\left({z}\right)\right){,}{\mathrm{arg}}{}\left({z}\right){=}{\mathrm{arctan}}{}\left({\mathrm{\Im }}{}\left({z}\right){,}{\mathrm{\Re }}{}\left({z}\right)\right){,}\left[{\mathrm{arg}}{}\left({z}{}{a}\right){=}{\mathrm{arg}}{}\left({z}\right){,}{0}{<}{a}\right]{,}{\mathrm{arg}}{}\left({z}{}{a}\right){=}{\mathrm{arg}}{}\left({z}\right){+}{\mathrm{arg}}{}\left({a}\right){+}{2}{}{\mathrm{\pi }}{}⌊\frac{{1}}{{2}}{-}\frac{{\mathrm{arg}}{}\left({z}\right)}{{2}{}{\mathrm{\pi }}}{-}\frac{{\mathrm{arg}}{}\left({a}\right)}{{2}{}{\mathrm{\pi }}}⌋{,}{\mathrm{arg}}{}\left(\frac{{z}}{{a}}\right){=}{\mathrm{arg}}{}\left({z}\right){-}{\mathrm{arg}}{}\left({a}\right){+}{2}{}{\mathrm{\pi }}{}⌊\frac{{1}}{{2}}{-}\frac{{\mathrm{arg}}{}\left({z}\right)}{{2}{}{\mathrm{\pi }}}{+}\frac{{\mathrm{arg}}{}\left({a}\right)}{{2}{}{\mathrm{\pi }}}⌋{,}\left[{\mathrm{arg}}{}\left({{z}}^{{a}}\right){=}{a}{}{\mathrm{arg}}{}\left({z}\right){,}{\mathrm{And}}{}\left({a}{::}{\mathrm{real}}{,}{-}{\mathrm{\pi }}{<}{a}{}{\mathrm{arg}}{}\left({z}\right){,}{a}{}{\mathrm{arg}}{}\left({z}\right){<}{\mathrm{\pi }}\right)\right]{,}\left[{\mathrm{arg}}{}\left({{z}}^{{a}}\right){=}{\mathrm{arg}}{}\left({{ⅇ}}^{{I}{}{a}{}{\mathrm{arg}}{}\left({z}\right)}\right){,}{a}{::}{\mathrm{real}}\right]{,}{\mathrm{arg}}{}\left({{z}}^{{a}}\right){=}{\mathrm{arctan}}{}\left({\mathrm{sin}}{}\left({\mathrm{arctan}}{}\left({\mathrm{\Im }}{}\left({z}\right){,}{\mathrm{\Re }}{}\left({z}\right)\right){}{\mathrm{\Re }}{}\left({a}\right){+}{\mathrm{\Im }}{}\left({a}\right){}{\mathrm{ln}}{}\left(\left|{z}\right|\right)\right){,}{\mathrm{cos}}{}\left({\mathrm{arctan}}{}\left({\mathrm{\Im }}{}\left({z}\right){,}{\mathrm{\Re }}{}\left({z}\right)\right){}{\mathrm{\Re }}{}\left({a}\right){+}{\mathrm{\Im }}{}\left({a}\right){}{\mathrm{ln}}{}\left(\left|{z}\right|\right)\right)\right)\right]$ (1.1.1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{Re}\right)$
 $\left[{\mathrm{\Re }}{}\left({I}{}{z}\right){=}{-}{\mathrm{\Im }}{}\left({z}\right){,}{\mathrm{\Re }}{}\left({z}\right){=}\frac{{z}{}\left({1}{+}\frac{{1}}{{{ⅇ}}^{{2}{}{I}{}{\mathrm{arg}}{}\left({z}\right)}}\right)}{{2}}{,}{\mathrm{\Re }}{}\left({z}\right){=}\frac{{z}{}\left({1}{+}\frac{{1}}{{{\mathrm{signum}}{}\left({z}\right)}^{{2}}}\right)}{{2}}{,}{\mathrm{\Re }}{}\left({z}\right){=}\frac{{z}}{{2}}{+}\frac{{\left|{z}\right|}^{{2}}}{{2}{}{z}}{,}{\mathrm{\Re }}{}\left({z}\right){=}\frac{{z}}{{2}}{+}\frac{\stackrel{{&conjugate0;}}{{z}}}{{2}}{,}\left[{\mathrm{\Re }}{}\left({z}{}{a}\right){=}{a}{}{\mathrm{\Re }}{}\left({z}\right){,}{a}{::}{\mathrm{real}}\right]{,}{\mathrm{\Re }}{}\left({z}{}{a}\right){=}{\mathrm{\Re }}{}\left({z}\right){}{\mathrm{\Re }}{}\left({a}\right){-}{\mathrm{\Im }}{}\left({a}\right){}{\mathrm{\Im }}{}\left({z}\right){,}{\mathrm{\Re }}{}\left(\frac{{z}}{{a}}\right){=}\frac{{\mathrm{\Re }}{}\left({z}\right){}{\mathrm{\Re }}{}\left({a}\right){+}{\mathrm{\Im }}{}\left({a}\right){}{\mathrm{\Im }}{}\left({z}\right)}{{\left|{a}\right|}^{{2}}}{,}{\mathrm{\Re }}{}\left({{z}}^{{a}}\right){=}{\left|{z}\right|}^{{\mathrm{\Re }}{}\left({a}\right)}{}{{ⅇ}}^{{-}{\mathrm{arg}}{}\left({z}\right){}{\mathrm{\Im }}{}\left({a}\right)}{}{\mathrm{cos}}{}\left({\mathrm{arg}}{}\left({z}\right){}{\mathrm{\Re }}{}\left({a}\right){+}{\mathrm{\Im }}{}\left({a}\right){}{\mathrm{ln}}{}\left(\left|{z}\right|\right)\right)\right]$ (1.1.2)
 • The Jacobi elliptic functions:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{JacobiAM}\right)$
 $\left[\left[{\mathrm{am}}{}\left({z}{|}{k}\right){=}{\mathrm{arccos}}{}\left({\mathrm{cn}}{}\left({z}{|}\right)\right)\right]\right]$