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

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{StruveL}\right)$
 $\left[{\mathrm{StruveL}}{}\left({a}{,}{z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({-}\frac{\frac{{1}}{{2}}{}{I}{}{\left({-}{1}\right)}^{\frac{{1}}{{2}}{}{a}{+}\frac{{1}}{{2}}{+}{2}{}{\mathrm{_k1}}}{}{{z}}^{{a}{+}{1}{+}{2}{}{\mathrm{_k1}}}}{{{ⅇ}}^{\frac{{1}}{{2}}{}{I}{}{a}{}{\mathrm{π}}}{}{{2}}^{{a}{+}{2}{}{\mathrm{_k1}}}{}{\mathrm{Γ}}{}\left(\frac{{3}}{{2}}{+}{a}{+}{\mathrm{_k1}}\right){}{\mathrm{Γ}}{}\left(\frac{{3}}{{2}}{+}{\mathrm{_k1}}\right)}\right){,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{+}\frac{{3}}{{2}}{::}{\mathbf{Not}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{nonnegint}}\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{JacobiTheta1}\left(a,z\right)\right)$
 $\left[{\mathrm{JacobiTheta1}}{}\left({a}{,}{z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{2}{}{{z}}^{{\left({\mathrm{_k1}}{+}\frac{{1}}{{2}}\right)}^{{2}}}{}{\mathrm{sin}}{}\left({a}{}\left({2}{}{\mathrm{_k1}}{+}{1}\right)\right){}{\left({-}{1}\right)}^{{\mathrm{_k1}}}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left|{z}\right|{<}{1}\right]$ (2)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{cos}\right)$
 $\left[{\mathrm{cos}}{}\left({z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{\mathrm{_k1}}}{}{{z}}^{{2}{}{\mathrm{_k1}}}}{\left({2}{}{\mathrm{_k1}}\right){!}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]$ (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.

 > $f≔\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{Stirling1}\left(n,z\right)\right)$
 ${f}{≔}\left[{\mathrm{Stirling1}}{}\left({n}{,}{z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{n}{-}{z}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{_k1}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{2}{}{\mathrm{_k1}}{-}{\mathrm{_k2}}}{}{\mathrm{binomial}}{}\left({n}{-}{1}{+}{\mathrm{_k1}}{,}{n}{-}{z}{+}{\mathrm{_k1}}\right){}{\mathrm{binomial}}{}\left({2}{}{n}{-}{z}{,}{n}{-}{z}{-}{\mathrm{_k1}}\right){}{\mathrm{binomial}}{}\left({\mathrm{_k1}}{,}{\mathrm{_k2}}\right){}{{\mathrm{_k2}}}^{{n}{-}{z}{+}{\mathrm{_k1}}}}{{\mathrm{_k1}}{!}}{,}{n}{::}{\mathrm{nonnegint}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{z}{::}{\mathrm{nonnegint}}\right]$ (5)
 > $\mathrm{depends}\left(f,n\right),\mathrm{depends}\left(f,z\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (6)
 >