FunctionAdvisor/special_values - Maple Programming Help

return the special values of a given mathematical function

 Calling Sequence FunctionAdvisor(special_values, math_function)

Parameters

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

Description

 • The FunctionAdvisor(special_values, math_function) command returns special values of the function computed at a selected list of special points.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{arccsc}\right)$
 $\left[{\mathrm{arccsc}}{}\left({-1}\right){=}{-}\frac{{\mathrm{\pi }}}{{2}}{,}{\mathrm{arccsc}}{}\left({-}\frac{{2}{}\sqrt{{3}}}{{3}}\right){=}{-}\frac{{\mathrm{\pi }}}{{3}}{,}{\mathrm{arccsc}}{}\left({-}\sqrt{{2}}\right){=}{-}\frac{{\mathrm{\pi }}}{{4}}{,}{\mathrm{arccsc}}{}\left({-2}\right){=}{-}\frac{{\mathrm{\pi }}}{{6}}{,}{\mathrm{arccsc}}{}\left({2}\right){=}\frac{{\mathrm{\pi }}}{{6}}{,}{\mathrm{arccsc}}{}\left(\sqrt{{2}}\right){=}\frac{{\mathrm{\pi }}}{{4}}{,}{\mathrm{arccsc}}{}\left(\frac{{2}{}\sqrt{{3}}}{{3}}\right){=}\frac{{\mathrm{\pi }}}{{3}}{,}{\mathrm{arccsc}}{}\left({1}\right){=}\frac{{\mathrm{\pi }}}{{2}}{,}{\mathrm{arccsc}}{}\left({I}\right){=}{-I}{}{\mathrm{ln}}{}\left({1}{+}\sqrt{{2}}\right){,}{\mathrm{arccsc}}{}\left({-I}\right){=}{I}{}{\mathrm{ln}}{}\left({1}{+}\sqrt{{2}}\right){,}{\mathrm{arccsc}}{}\left({0}\right){=}\frac{{\mathrm{\pi }}}{{2}}{-}{\mathrm{\infty }}{}{I}{,}{\mathrm{arccsc}}{}\left({\mathrm{\infty }}\right){=}{0}{,}{\mathrm{arccsc}}{}\left({-}{\mathrm{\infty }}\right){=}{0}\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{sin}\right)$
 $\left[{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{6}}\right){=}\frac{{1}}{{2}}{,}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{4}}\right){=}\frac{\sqrt{{2}}}{{2}}{,}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{3}}\right){=}\frac{\sqrt{{3}}}{{2}}{,}{\mathrm{sin}}{}\left({\mathrm{\infty }}\right){=}{\mathrm{undefined}}{,}{\mathrm{sin}}{}\left({\mathrm{\infty }}{}{I}\right){=}{\mathrm{\infty }}{}{I}{,}\left[{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{n}\right){=}{0}{,}{n}{::}{'}{\mathrm{integer}}{'}\right]{,}\left[{\mathrm{sin}}{}\left(\frac{\left({2}{}{n}{+}{1}\right){}{\mathrm{\pi }}}{{2}}\right){=}{-1}{,}{n}{::}{\mathrm{odd}}\right]{,}\left[{\mathrm{sin}}{}\left(\frac{\left({2}{}{n}{+}{1}\right){}{\mathrm{\pi }}}{{2}}\right){=}{1}{,}{n}{::}{\mathrm{even}}\right]\right]$ (2)
 > $\mathrm{ex1}≔\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{binomial}\right)$
 ${\mathrm{ex1}}{≔}\left[\left(\genfrac{}{}{0}{}{{0}}{{z}}\right){=}\left\{\begin{array}{cc}{1}& {z}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{z}\right)}{{\mathrm{\pi }}{}{z}}& {\mathrm{otherwise}}\end{array}\right\{,}\left(\genfrac{}{}{0}{}{{a}}{{0}}\right){=}{1}{,}\left(\genfrac{}{}{0}{}{{a}}{{1}}\right){=}{a}{,}\left(\genfrac{}{}{0}{}{{0}}{{0}}\right){=}{1}\right]$ (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 a or z.

 > $\mathrm{depends}\left(\left[\mathrm{ex1}\right],a\right),\mathrm{depends}\left(\left[\mathrm{ex1}\right],z\right)$
 ${\mathrm{false}}{,}{\mathrm{false}}$ (4)

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

 > $\mathrm{ex2}≔\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{binomial}\left(a,z\right)\right)$
 ${\mathrm{ex2}}{≔}\left[\left(\genfrac{}{}{0}{}{{0}}{{z}}\right){=}\left\{\begin{array}{cc}{1}& {z}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{z}\right)}{{\mathrm{\pi }}{}{z}}& {\mathrm{otherwise}}\end{array}\right\{,}\left(\genfrac{}{}{0}{}{{a}}{{0}}\right){=}{1}{,}\left(\genfrac{}{}{0}{}{{a}}{{1}}\right){=}{a}{,}\left(\genfrac{}{}{0}{}{{0}}{{0}}\right){=}{1}\right]$ (5)
 > $\mathrm{depends}\left(\left[\mathrm{ex2}\right],a\right),\mathrm{depends}\left(\left[\mathrm{ex2}\right],z\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (6)

For functions which accept more than one argument, the special values of interest could be restricted by passing the function call. For example, these are special values for $\mathrm{\Psi }\left(1,z\right)$

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{\Psi }\left(1,z\right)\right)$
 $\left[{\mathrm{\Psi }}{}\left({-1}{,}{z}\right){=}{\mathrm{lnGAMMA}}{}\left({z}\right){-}\frac{{\mathrm{ln}}{}\left({2}{}{\mathrm{\pi }}\right)}{{2}}{,}{\mathrm{\Psi }}{}\left({1}{,}{1}\right){=}\frac{{{\mathrm{\pi }}}^{{2}}}{{6}}{,}{\mathrm{\Psi }}{}\left({1}{,}\frac{{1}}{{2}}\right){=}\frac{{{\mathrm{\pi }}}^{{2}}}{{2}}{,}{\mathrm{\Psi }}{}\left({1}{,}\frac{{1}}{{4}}\right){=}{{\mathrm{\pi }}}^{{2}}{+}{8}{}{\mathrm{Catalan}}{,}{\mathrm{\Psi }}{}\left({0}{,}{z}\right){=}{\mathrm{\Psi }}{}\left({z}\right){,}{\mathrm{\Psi }}{}\left({1}{,}{-1}\right){=}{\mathrm{\infty }}{,}{\mathrm{\Psi }}{}\left({1}{,}{0}\right){=}{\mathrm{\infty }}{,}\left[{\mathrm{\Psi }}{}\left({1}{,}{-}{n}\right){=}{\mathrm{\infty }}{,}{\mathrm{\Psi }}{}\left({1}{,}{n}\right){=}\frac{{{\mathrm{\pi }}}^{{2}}}{{6}}{-}\left({\sum }_{{k}{=}{1}}^{{n}{-}{1}}{}\frac{{1}}{{{k}}^{{2}}}\right){,}{\mathrm{\Psi }}{}\left({1}{,}{n}{+}\frac{{1}}{{2}}\right){=}\frac{{{\mathrm{\pi }}}^{{2}}}{{2}}{-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{4}}{{\left({2}{}{k}{-}{1}\right)}^{{2}}}\right){,}{\mathrm{\Psi }}{}\left({1}{,}\frac{{1}}{{2}}{-}{n}\right){=}\frac{{{\mathrm{\pi }}}^{{2}}}{{2}}{+}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{4}}{{\left({2}{}{k}{-}{1}\right)}^{{2}}}\right){,}{n}{::}{'}{\mathrm{posint}}{'}\right]\right]$ (7)

and these are special values for $\mathrm{\Psi }\left(z\right)$.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{\Psi }\left(z\right)\right)$
 $\left[{\mathrm{\Psi }}{}\left({-}\frac{{3}}{{2}}\right){=}\frac{{8}}{{3}}{-}{\mathrm{\gamma }}{-}{2}{}{\mathrm{ln}}{}\left({2}\right){,}{\mathrm{\Psi }}{}\left({-}\frac{{1}}{{2}}\right){=}{2}{-}{\mathrm{\gamma }}{-}{2}{}{\mathrm{ln}}{}\left({2}\right){,}{\mathrm{\Psi }}{}\left(\frac{{1}}{{4}}\right){=}{-}{\mathrm{\gamma }}{-}{3}{}{\mathrm{ln}}{}\left({2}\right){-}\frac{{\mathrm{\pi }}}{{2}}{,}{\mathrm{\Psi }}{}\left(\frac{{1}}{{2}}\right){=}{-}{\mathrm{\gamma }}{-}{2}{}{\mathrm{ln}}{}\left({2}\right){,}{\mathrm{\Psi }}{}\left(\frac{{3}}{{4}}\right){=}{-}{\mathrm{\gamma }}{-}{3}{}{\mathrm{ln}}{}\left({2}\right){+}\frac{{\mathrm{\pi }}}{{2}}{,}{\mathrm{\Psi }}{}\left(\frac{{1}}{{3}}\right){=}{-}{\mathrm{\gamma }}{-}\frac{{\mathrm{\pi }}{}\sqrt{{3}}}{{6}}{-}\frac{{3}{}{\mathrm{ln}}{}\left({3}\right)}{{2}}{,}{\mathrm{\Psi }}{}\left(\frac{{2}}{{3}}\right){=}{-}{\mathrm{\gamma }}{+}\frac{{\mathrm{\pi }}{}\sqrt{{3}}}{{6}}{-}\frac{{3}{}{\mathrm{ln}}{}\left({3}\right)}{{2}}{,}{\mathrm{\Psi }}{}\left({1}\right){=}{-}{\mathrm{\gamma }}{,}{\mathrm{\Psi }}{}\left({2}\right){=}{1}{-}{\mathrm{\gamma }}{,}{\mathrm{\Psi }}{}\left({-1}\right){=}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}{,}{\mathrm{\Psi }}{}\left({0}\right){=}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}{,}{\mathrm{\Psi }}{}\left({\mathrm{\infty }}\right){=}{\mathrm{\infty }}{,}{\mathrm{\Psi }}{}\left({-}{\mathrm{\infty }}\right){=}{\mathrm{undefined}}{,}{\mathrm{\Psi }}{}\left({\mathrm{\infty }}{}{I}\right){=}{\mathrm{\infty }}{+}\frac{{I}{}{\mathrm{\pi }}}{{2}}{,}{\mathrm{\Psi }}{}\left({-}{\mathrm{\infty }}{}{I}\right){=}{\mathrm{\infty }}{-}\frac{{I}{}{\mathrm{\pi }}}{{2}}{,}\left[{\mathrm{\Psi }}{}\left({n}\right){=}\left({\sum }_{{k}{=}{1}}^{{n}{-}{1}}{}\frac{{1}}{{k}}\right){-}{\mathrm{\gamma }}{,}{n}{::}{'}{\mathrm{posint}}{'}\right]{,}\left[{\mathrm{\Psi }}{}\left({n}{+}\frac{{p}}{{q}}\right){=}{q}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{{1}}{{k}{}{q}{+}{p}}\right){+}{2}{}\left({\sum }_{{k}{=}{1}}^{⌊\frac{{q}}{{2}}{-}\frac{{1}}{{2}}⌋}{}{\mathrm{cos}}{}\left(\frac{{2}{}{\mathrm{\pi }}{}{p}{}{k}}{{q}}\right){}{\mathrm{ln}}{}\left({\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}{k}}{{q}}\right)\right)\right){-}\frac{{\mathrm{\pi }}{}{\mathrm{cot}}{}\left(\frac{{\mathrm{\pi }}{}{p}}{{q}}\right)}{{2}}{-}{\mathrm{ln}}{}\left({2}{}{q}\right){-}{\mathrm{\gamma }}{,}{\mathrm{\Psi }}{}\left(\frac{{p}}{{q}}{-}{n}\right){=}{q}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{{1}}{{q}{}\left({k}{+}{1}\right){-}{p}}\right){+}{2}{}\left({\sum }_{{k}{=}{1}}^{⌊\frac{{q}}{{2}}{-}\frac{{1}}{{2}}⌋}{}{\mathrm{cos}}{}\left(\frac{{2}{}{\mathrm{\pi }}{}{p}{}{k}}{{q}}\right){}{\mathrm{ln}}{}\left({\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}{k}}{{q}}\right)\right)\right){-}\frac{{\mathrm{\pi }}{}{\mathrm{cot}}{}\left(\frac{{\mathrm{\pi }}{}{p}}{{q}}\right)}{{2}}{-}{\mathrm{ln}}{}\left({2}{}{q}\right){-}{\mathrm{\gamma }}{,}{n}{::}{'}{\mathrm{nonnegint}}{'}{\wedge }\left\{{p}{,}{q}\right\}{::}{\mathrm{set}}{}\left({'}{\mathrm{posint}}{'}\right){\wedge }{p}{<}{q}\right]\right]$ (8)
 > 

In these cases, when the FunctionAdvisor command is called with the function name, for example, Psi, the values are listed on the screen starting with those involving less arguments in the function call (in this example, $\mathrm{\Psi }\left(z\right)$).