return the special values of a given mathematical 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

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{arccsc}\right)$
 $\left[{\mathrm{arccsc}}{}\left({-}{1}\right){=}{-}\frac{{1}}{{2}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left({-}\frac{{2}}{{3}}{}\sqrt{{3}}\right){=}{-}\frac{{1}}{{3}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left({-}\sqrt{{2}}\right){=}{-}\frac{{1}}{{4}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left({-}{2}\right){=}{-}\frac{{1}}{{6}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left({2}\right){=}\frac{{1}}{{6}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left(\sqrt{{2}}\right){=}\frac{{1}}{{4}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left(\frac{{2}}{{3}}{}\sqrt{{3}}\right){=}\frac{{1}}{{3}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left({1}\right){=}\frac{{1}}{{2}}{}{\mathrm{π}}{,}{\mathrm{arccsc}}{}\left({I}\right){=}{-}{I}{}{\mathrm{ln}}{}\left({1}{+}\sqrt{{2}}\right){,}{\mathrm{arccsc}}{}\left({-}{I}\right){=}{I}{}{\mathrm{ln}}{}\left({1}{+}\sqrt{{2}}\right){,}{\mathrm{arccsc}}{}\left({0}\right){=}\frac{{1}}{{2}}{}{\mathrm{π}}{-}{\mathrm{∞}}{}{I}{,}{\mathrm{arccsc}}{}\left({\mathrm{∞}}\right){=}{0}{,}{\mathrm{arccsc}}{}\left({-}{\mathrm{∞}}\right){=}{0}\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{sin}\right)$
 $\left[{\mathrm{sin}}{}\left(\frac{{1}}{{6}}{}{\mathrm{π}}\right){=}\frac{{1}}{{2}}{,}{\mathrm{sin}}{}\left(\frac{{1}}{{4}}{}{\mathrm{π}}\right){=}\frac{{1}}{{2}}{}\sqrt{{2}}{,}{\mathrm{sin}}{}\left(\frac{{1}}{{3}}{}{\mathrm{π}}\right){=}\frac{{1}}{{2}}{}\sqrt{{3}}{,}{\mathrm{sin}}{}\left({\mathrm{∞}}\right){=}{\mathrm{undefined}}{,}{\mathrm{sin}}{}\left({\mathrm{∞}}{}{I}\right){=}{\mathrm{∞}}{}{I}{,}\left[{\mathrm{sin}}{}\left({\mathrm{π}}{}{n}\right){=}{0}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{integer}}\right]{,}\left[{\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}\left({2}{}{n}{+}{1}\right){}{\mathrm{π}}\right){=}{-}{1}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{odd}}\right]{,}\left[{\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}\left({2}{}{n}{+}{1}\right){}{\mathrm{π}}\right){=}{1}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{even}}\right]\right]$ (2)
 > $\mathrm{ex1}≔\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{binomial}\right)$
 ${\mathrm{ex1}}{≔}\left[{\mathrm{binomial}}{}\left({0}{,}{z}\right){=}{{}\begin{array}{cc}{1}& {z}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{π}}{}{z}\right)}{{\mathrm{π}}{}{z}}& {\mathrm{otherwise}}\end{array}{,}{\mathrm{binomial}}{}\left({a}{,}{0}\right){=}{1}{,}{\mathrm{binomial}}{}\left({a}{,}{1}\right){=}{a}{,}{\mathrm{binomial}}{}\left({0}{,}{0}\right){=}{1}\right]$ (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.

 > $\mathrm{ex2}≔\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{binomial}\left(a,z\right)\right)$
 ${\mathrm{ex2}}{≔}\left[{\mathrm{binomial}}{}\left({0}{,}{z}\right){=}{{}\begin{array}{cc}{1}& {z}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{π}}{}{z}\right)}{{\mathrm{π}}{}{z}}& {\mathrm{otherwise}}\end{array}{,}{\mathrm{binomial}}{}\left({a}{,}{0}\right){=}{1}{,}{\mathrm{binomial}}{}\left({a}{,}{1}\right){=}{a}{,}{\mathrm{binomial}}{}\left({0}{,}{0}\right){=}{1}\right]$ (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 $\mathrm{Psi}\left(1,z\right)$

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{Ψ}\left(1,z\right)\right)$
 $\left[{\mathrm{Ψ}}{}\left({-}{1}{,}{z}\right){=}{\mathrm{lnGAMMA}}{}\left({z}\right){-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({2}{}{\mathrm{π}}\right){,}{\mathrm{Ψ}}{}\left({1}{,}{1}\right){=}\frac{{1}}{{6}}{}{{\mathrm{π}}}^{{2}}{,}{\mathrm{Ψ}}{}\left({1}{,}\frac{{1}}{{2}}\right){=}\frac{{1}}{{2}}{}{{\mathrm{π}}}^{{2}}{,}{\mathrm{Ψ}}{}\left({1}{,}\frac{{1}}{{4}}\right){=}{{\mathrm{π}}}^{{2}}{+}{8}{}{\mathrm{Catalan}}{,}{\mathrm{Ψ}}{}\left({0}{,}{z}\right){=}{\mathrm{Ψ}}{}\left({z}\right){,}{\mathrm{Ψ}}{}\left({1}{,}{-}{1}\right){=}{\mathrm{∞}}{,}{\mathrm{Ψ}}{}\left({1}{,}{0}\right){=}{\mathrm{∞}}{,}\left[{\mathrm{Ψ}}{}\left({1}{,}{-}{n}\right){=}{\mathrm{∞}}{,}{\mathrm{Ψ}}{}\left({1}{,}{n}\right){=}\frac{{1}}{{6}}{}{{\mathrm{π}}}^{{2}}{-}{\sum }_{{k}{=}{1}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{{k}}^{{2}}}{,}{\mathrm{Ψ}}{}\left({1}{,}{n}{+}\frac{{1}}{{2}}\right){=}\frac{{1}}{{2}}{}{{\mathrm{π}}}^{{2}}{-}{\sum }_{{k}{=}{1}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{4}}{{\left({2}{}{k}{-}{1}\right)}^{{2}}}{,}{\mathrm{Ψ}}{}\left({1}{,}\frac{{1}}{{2}}{-}{n}\right){=}\frac{{1}}{{2}}{}{{\mathrm{π}}}^{{2}}{+}{\sum }_{{k}{=}{1}}^{{n}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{4}}{{\left({2}{}{k}{-}{1}\right)}^{{2}}}{,}{n}{::}{\mathrm{posint}}\right]\right]$ (7)

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

 > $\mathrm{FunctionAdvisor}\left(\mathrm{special_values},\mathrm{Ψ}\left(z\right)\right)$
 $\left[{\mathrm{Ψ}}{}\left({-}\frac{{3}}{{2}}\right){=}\frac{{8}}{{3}}{-}{\mathrm{γ}}{-}{2}{}{\mathrm{ln}}{}\left({2}\right){,}{\mathrm{Ψ}}{}\left({-}\frac{{1}}{{2}}\right){=}{2}{-}{\mathrm{γ}}{-}{2}{}{\mathrm{ln}}{}\left({2}\right){,}{\mathrm{Ψ}}{}\left(\frac{{1}}{{4}}\right){=}{-}{\mathrm{γ}}{-}{3}{}{\mathrm{ln}}{}\left({2}\right){-}\frac{{1}}{{2}}{}{\mathrm{π}}{,}{\mathrm{Ψ}}{}\left(\frac{{1}}{{2}}\right){=}{-}{\mathrm{γ}}{-}{2}{}{\mathrm{ln}}{}\left({2}\right){,}{\mathrm{Ψ}}{}\left(\frac{{3}}{{4}}\right){=}{-}{\mathrm{γ}}{-}{3}{}{\mathrm{ln}}{}\left({2}\right){+}\frac{{1}}{{2}}{}{\mathrm{π}}{,}{\mathrm{Ψ}}{}\left(\frac{{1}}{{3}}\right){=}{-}{\mathrm{γ}}{-}\frac{{1}}{{6}}{}{\mathrm{π}}{}\sqrt{{3}}{-}\frac{{3}}{{2}}{}{\mathrm{ln}}{}\left({3}\right){,}{\mathrm{Ψ}}{}\left(\frac{{2}}{{3}}\right){=}{-}{\mathrm{γ}}{+}\frac{{1}}{{6}}{}{\mathrm{π}}{}\sqrt{{3}}{-}\frac{{3}}{{2}}{}{\mathrm{ln}}{}\left({3}\right){,}{\mathrm{Ψ}}{}\left({1}\right){=}{-}{\mathrm{γ}}{,}{\mathrm{Ψ}}{}\left({2}\right){=}{1}{-}{\mathrm{γ}}{,}{\mathrm{Ψ}}{}\left({-}{1}\right){=}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}{,}{\mathrm{Ψ}}{}\left({0}\right){=}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}{,}{\mathrm{Ψ}}{}\left({\mathrm{∞}}\right){=}{\mathrm{∞}}{,}{\mathrm{Ψ}}{}\left({-}{\mathrm{∞}}\right){=}{\mathrm{undefined}}{,}{\mathrm{Ψ}}{}\left({\mathrm{∞}}{}{I}\right){=}{\mathrm{∞}}{+}\frac{{1}}{{2}}{}{I}{}{\mathrm{π}}{,}{\mathrm{Ψ}}{}\left({-}{\mathrm{∞}}{}{I}\right){=}{\mathrm{∞}}{-}\frac{{1}}{{2}}{}{I}{}{\mathrm{π}}{,}\left[{\mathrm{Ψ}}{}\left({n}\right){=}{\sum }_{{k}{=}{1}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{k}}{-}{\mathrm{γ}}{,}{n}{::}{\mathrm{posint}}\right]{,}\left[{\mathrm{Ψ}}{}\left({n}{+}\frac{{p}}{{q}}\right){=}{q}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{k}{}{q}{+}{p}}\right){+}{2}{}\left({\sum }_{{k}{=}{1}}^{{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{}{q}{-}\frac{{1}}{{2}}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\mathrm{cos}}{}\left(\frac{{2}{}{\mathrm{π}}{}{p}{}{k}}{{q}}\right){}{\mathrm{ln}}{}\left({\mathrm{sin}}{}\left(\frac{{\mathrm{π}}{}{k}}{{q}}\right)\right)\right){-}\frac{{1}}{{2}}{}{\mathrm{π}}{}{\mathrm{cot}}{}\left(\frac{{\mathrm{π}}{}{p}}{{q}}\right){-}{\mathrm{ln}}{}\left({2}{}{q}\right){-}{\mathrm{γ}}{,}{\mathrm{Ψ}}{}\left(\frac{{p}}{{q}}{-}{n}\right){=}{q}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{1}}{{q}{}\left({k}{+}{1}\right){-}{p}}\right){+}{2}{}\left({\sum }_{{k}{=}{1}}^{{\mathrm{floor}}{}\left(\frac{{1}}{{2}}{}{q}{-}\frac{{1}}{{2}}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\mathrm{cos}}{}\left(\frac{{2}{}{\mathrm{π}}{}{p}{}{k}}{{q}}\right){}{\mathrm{ln}}{}\left({\mathrm{sin}}{}\left(\frac{{\mathrm{π}}{}{k}}{{q}}\right)\right)\right){-}\frac{{1}}{{2}}{}{\mathrm{π}}{}{\mathrm{cot}}{}\left(\frac{{\mathrm{π}}{}{p}}{{q}}\right){-}{\mathrm{ln}}{}\left({2}{}{q}\right){-}{\mathrm{γ}}{,}{n}{::}{\mathrm{nonnegint}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{{p}{,}{q}\right\}{::}{\mathrm{set}}{}\left({\mathrm{posint}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{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, $\mathrm{Psi}\left(z\right)$).