asymptotic_expansion - Maple Help

return the asymptotic expansion of a given mathematical function

Parameters

 asymptotic_expansion - literal name; 'asymptotic_expansion' math_function - Maple name of mathematical function

Description

 • The FunctionAdvisor(asymptotic_expansion, math_function) command returns the asymptotic expansion of the function, if possible.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{asymptotic_expansion},\mathrm{arcsin}\right)$
 ${\mathrm{asympt}}{}\left({\mathrm{arcsin}}{}\left({z}\right){,}{z}{,}{4}\right){=}{-}{I}{}\left({\mathrm{ln}}{}\left({2}\right){+}\frac{{I}{}{\mathrm{\pi }}}{{2}}{+}{\mathrm{ln}}{}\left({z}\right)\right){+}\frac{{I}}{{4}{}{{z}}^{{2}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{z}}^{{4}}}\right)$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{asymptotic_expansion},\mathrm{erf}\right)$
 ${\mathrm{asympt}}{}\left({\mathrm{erf}}{}\left({z}\right){,}{z}{,}{4}\right){=}{1}{+}\frac{{-}\frac{{1}}{\sqrt{{\mathrm{\pi }}}{}{z}}{+}\frac{{1}}{{2}{}\sqrt{{\mathrm{\pi }}}{}{{z}}^{{3}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{z}}^{{5}}}\right)}{{{ⅇ}}^{{{z}}^{{2}}}}$ (2)

The variables used by the FunctionAdvisor command to create the calling sequence are local variables. To make the FunctionAdvisor command use global variables, pass the actual function call instead of the function name. Compare the following two input and output groups.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{asymptotic_expansion},\mathrm{FresnelS}\right)$
 ${\mathrm{asympt}}{}\left({\mathrm{FresnelS}}{}\left({z}\right){,}{z}{,}{4}\right){=}\frac{{1}}{{2}}{-}\frac{{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}{}{{z}}^{{2}}}{{2}}\right)}{{\mathrm{\pi }}{}{z}}{-}\frac{{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}{{z}}^{{2}}}{{2}}\right)}{{{\mathrm{\pi }}}^{{2}}{}{{z}}^{{3}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{z}}^{{5}}}\right)$ (3)
 > $\mathrm{has}\left(,z\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{asymptotic_expansion},\mathrm{FresnelS}\left(z\right)\right)$
 ${\mathrm{asympt}}{}\left({\mathrm{FresnelS}}{}\left({z}\right){,}{z}{,}{4}\right){=}\frac{{1}}{{2}}{-}\frac{{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}{}{{z}}^{{2}}}{{2}}\right)}{{\mathrm{\pi }}{}{z}}{-}\frac{{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}{{z}}^{{2}}}{{2}}\right)}{{{\mathrm{\pi }}}^{{2}}{}{{z}}^{{3}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{z}}^{{5}}}\right)$ (5)
 > $\mathrm{has}\left(,z\right)$
 ${\mathrm{true}}$ (6)