sum_form - Maple Help

return the sum form of a given mathematical function

Parameters

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

Description

 • The FunctionAdvisor(sum_form, math_function) command returns the sum form of the function if it exists.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{StruveL}\right)$
 $\left[{\mathrm{StruveL}}{}\left({a}{,}{z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{-}\frac{{I}}{{2}}{}{\left({-1}\right)}^{\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{+}{2}{}{\mathrm{_k1}}}{}{{z}}^{{a}{+}{1}{+}{2}{}{\mathrm{_k1}}}}{{{ⅇ}}^{\frac{{I}}{{2}}{}{a}{}{\mathrm{\pi }}}{}{{2}}^{{a}{+}{2}{}{\mathrm{_k1}}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{\mathrm{_k1}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}{+}{\mathrm{_k1}}\right)}{,}\left({a}{+}\frac{{3}}{{2}}\right){::}\left({¬}{\mathrm{nonnegint}}\right)\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{JacobiTheta1}\left(a,z\right)\right)$
 $\left[{\mathrm{JacobiTheta1}}{}\left({a}{,}{z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}{2}{}{{z}}^{{\left({\mathrm{_k1}}{+}\frac{{1}}{{2}}\right)}^{{2}}}{}{\mathrm{sin}}{}\left({a}{}\left({2}{}{\mathrm{_k1}}{+}{1}\right)\right){}{\left({-1}\right)}^{{\mathrm{_k1}}}{,}\left|{z}\right|{<}{1}\right]$ (2)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{cos}\right)$
 $\left[{\mathrm{cos}}{}\left({z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-1}\right)}^{{\mathrm{_k1}}}{}{{z}}^{{2}{}{\mathrm{_k1}}}}{\left({2}{}{\mathrm{_k1}}\right){!}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]$ (3)

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.

 > $\mathrm{depends}\left(\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{cos}\right),z\right)$
 ${\mathrm{false}}$ (4)

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

 > $f≔\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{Stirling1}\left(n,z\right)\right)$
 ${f}{≔}\left[{\mathrm{Stirling1}}{}\left({n}{,}{z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}{-}{z}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{_k1}}}{}\frac{{\left({-1}\right)}^{{2}{}{\mathrm{_k1}}{-}{\mathrm{_k2}}}{}\left(\genfrac{}{}{0}{}{{n}{-}{1}{+}{\mathrm{_k1}}}{{n}{-}{z}{+}{\mathrm{_k1}}}\right){}\left(\genfrac{}{}{0}{}{{2}{}{n}{-}{z}}{{n}{-}{z}{-}{\mathrm{_k1}}}\right){}\left(\genfrac{}{}{0}{}{{\mathrm{_k1}}}{{\mathrm{_k2}}}\right){}{{\mathrm{_k2}}}^{{n}{-}{z}{+}{\mathrm{_k1}}}}{{\mathrm{_k1}}{!}}{,}{n}{::}{\mathrm{nonnegint}}{\wedge }{z}{::}{\mathrm{nonnegint}}\right]$ (5)
 > $\mathrm{depends}\left(f,n\right),\mathrm{depends}\left(f,z\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (6)