series - Maple Help

return the series expansion of a given mathematical function

Parameters

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

Description

 • The FunctionAdvisor(series, math_function) command returns the series expansion of the function when the system can compute it. Otherwise, the command returns NULL.
 • The order of the series expansion in the output is set to four. On the left-hand side there is the corresponding function call to compute the series in Maple to facilitate copy and paste. If you change the order in the calling sequence, it returns the number of terms requested.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{series},\mathrm{JacobiTheta3}\right)$
 ${\mathrm{series}}{}\left({\mathrm{JacobiTheta3}}{}\left({z}{,}{q}\right){,}{z}{,}{4}\right){=}{2}{}\left({\sum }_{{\mathrm{_k1}}{=}{1}}^{{\mathrm{\infty }}}{}{{q}}^{{{\mathrm{_k1}}}^{{2}}}\right){+}{1}{-}{4}{}\left({\sum }_{{\mathrm{_k1}}{=}{1}}^{{\mathrm{\infty }}}{}{{q}}^{{{\mathrm{_k1}}}^{{2}}}{}{{\mathrm{_k1}}}^{{2}}\right){}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{4}}\right){,}{\mathrm{series}}{}\left({\mathrm{JacobiTheta3}}{}\left({z}{,}{q}\right){,}{q}{,}{4}\right){=}{1}{+}{2}{}{\mathrm{cos}}{}\left({2}{}{z}\right){}{q}{+}{O}{}\left({{q}}^{{4}}\right)$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{series},\mathrm{sin}\right)$
 ${\mathrm{series}}{}\left({\mathrm{sin}}{}\left({z}\right){,}{z}{,}{4}\right){=}{z}{-}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}{O}{}\left({{z}}^{{5}}\right)$ (2)

The variables used by the FunctionAdvisor command to create the function calling sequences are local variables. Therefore, the previous example does not depend on z.

 > $\mathrm{depends}\left(,z\right)$
 ${\mathrm{false}}$ (3)

To make the FunctionAdvisor command return results using global variables, pass the function call itself.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{series},\mathrm{sin}\left(z\right)\right)$
 ${\mathrm{series}}{}\left({\mathrm{sin}}{}\left({z}\right){,}{z}{,}{4}\right){=}{z}{-}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}{O}{}\left({{z}}^{{5}}\right)$ (4)
 > $\mathrm{depends}\left(,z\right)$
 ${\mathrm{true}}$ (5)

Note that for some functions Maple knows how to compute the series expansion with respect to more than one function parameter. In these cases, a sequence of series expansions is returned.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{series},\mathrm{Β}\right)$
 ${\mathrm{series}}{}\left({\mathrm{Β}}{}\left({x}{,}{y}\right){,}{x}{,}{4}\right){=}{{x}}^{{-1}}{+}{-}{\mathrm{\Psi }}{}\left({y}\right){-}{\mathrm{\gamma }}{+}\left(\frac{{{\mathrm{\pi }}}^{{2}}}{{12}}{+}\frac{{{\mathrm{\gamma }}}^{{2}}}{{2}}{-}\frac{{\mathrm{\Psi }}{}\left({1}{,}{y}\right)}{{2}}{+}\frac{{{\mathrm{\Psi }}{}\left({y}\right)}^{{2}}}{{2}}{+}{\mathrm{\Psi }}{}\left({y}\right){}{\mathrm{\gamma }}\right){}{x}{+}\left({-}\frac{{\mathrm{\zeta }}{}\left({3}\right)}{{3}}{-}\frac{{{\mathrm{\pi }}}^{{2}}{}{\mathrm{\gamma }}}{{12}}{-}\frac{{{\mathrm{\gamma }}}^{{3}}}{{6}}{-}\frac{{\mathrm{\Psi }}{}\left({2}{,}{y}\right)}{{6}}{+}\frac{{\mathrm{\Psi }}{}\left({1}{,}{y}\right){}{\mathrm{\Psi }}{}\left({y}\right)}{{2}}{-}\frac{{{\mathrm{\Psi }}{}\left({y}\right)}^{{3}}}{{6}}{+}\frac{{\mathrm{\gamma }}{}{\mathrm{\Psi }}{}\left({1}{,}{y}\right)}{{2}}{-}\frac{{\mathrm{\gamma }}{}{{\mathrm{\Psi }}{}\left({y}\right)}^{{2}}}{{2}}{-}\frac{{\mathrm{\Psi }}{}\left({y}\right){}{{\mathrm{\gamma }}}^{{2}}}{{2}}{-}\frac{{\mathrm{\Psi }}{}\left({y}\right){}{{\mathrm{\pi }}}^{{2}}}{{12}}\right){}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right)$ (6)