return the identities of a given mathematical function

Parameters

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

Description

 • The FunctionAdvisor(identities, math_function) command returns a list of identities for that function.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{sin}\right)$
 $\left[{\mathrm{sin}}{}\left({\mathrm{arcsin}}{}\left({z}\right)\right){=}{z}{,}{\mathrm{sin}}{}\left({z}\right){=}{-}{\mathrm{sin}}{}\left({-}{z}\right){,}{\mathrm{sin}}{}\left({z}\right){=}{2}{}{\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}{z}\right){}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}{z}\right){,}{\mathrm{sin}}{}\left({z}\right){=}\frac{{1}}{{\mathrm{csc}}{}\left({z}\right)}{,}{\mathrm{sin}}{}\left({z}\right){=}\frac{{2}{}{\mathrm{tan}}{}\left(\frac{{1}}{{2}}{}{z}\right)}{{1}{+}{{\mathrm{tan}}{}\left(\frac{{1}}{{2}}{}{z}\right)}^{{2}}}{,}{\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{1}}{{2}}{}{I}{}\left({{ⅇ}}^{{I}{}{z}}{-}{{ⅇ}}^{{-}{I}{}{z}}\right){,}{{\mathrm{sin}}{}\left({z}\right)}^{{2}}{=}{1}{-}{{\mathrm{cos}}{}\left({z}\right)}^{{2}}{,}{{\mathrm{sin}}{}\left({z}\right)}^{{2}}{=}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{\mathrm{cos}}{}\left({2}{}{z}\right)\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{describe},\mathrm{coth}\right)$
 ${\mathrm{coth}}{=}{\mathrm{hyperbolic cotangent}}$ (2)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{coth}\right)$
 $\left[{\mathrm{coth}}{}\left({z}\right){=}\frac{{\left({{ⅇ}}^{{z}}\right)}^{{2}}{+}{1}}{{\left({{ⅇ}}^{{z}}\right)}^{{2}}{-}{1}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]$ (3)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{coth}\right)$
 $\left[{\mathrm{coth}}{}\left({z}\right){=}{-}{\mathrm{coth}}{}\left({-}{z}\right){,}{\mathrm{coth}}{}\left({z}\right){=}{-}\frac{{1}}{{2}}{}\frac{{-}{{\mathrm{coth}}{}\left(\frac{{1}}{{2}}{}{z}\right)}^{{2}}{-}{1}}{{\mathrm{coth}}{}\left(\frac{{1}}{{2}}{}{z}\right)}{,}{\mathrm{coth}}{}\left({z}\right){=}\frac{{\mathrm{cosh}}{}\left({z}\right)}{{\mathrm{sinh}}{}\left({z}\right)}{,}{\mathrm{coth}}{}\left({z}\right){=}\frac{{1}{+}{\mathrm{cosh}}{}\left({2}{}{z}\right)}{{\mathrm{sinh}}{}\left({2}{}{z}\right)}{,}{\mathrm{coth}}{}\left({z}\right){=}{-}\frac{{\mathrm{sinh}}{}\left({2}{}{z}\right)}{{1}{-}{\mathrm{cosh}}{}\left({2}{}{z}\right)}{,}{\mathrm{coth}}{}\left({z}\right){=}\frac{{1}}{{\mathrm{tanh}}{}\left({z}\right)}{,}{\mathrm{coth}}{}\left({z}\right){=}\frac{{1}}{{2}}{}\frac{{1}{+}{{\mathrm{tanh}}{}\left(\frac{{1}}{{2}}{}{z}\right)}^{{2}}}{{\mathrm{tanh}}{}\left(\frac{{1}}{{2}}{}{z}\right)}{,}{\mathrm{coth}}{}\left({z}\right){=}{I}{}\left({-}\frac{{1}}{{2}}{}{I}{}{\mathrm{coth}}{}\left(\frac{{1}}{{2}}{}{z}\right){-}\frac{{1}}{{2}}{}{I}{}{\mathrm{tanh}}{}\left(\frac{{1}}{{2}}{}{z}\right)\right){,}{\mathrm{coth}}{}\left({z}\right){=}{-}\frac{{{ⅇ}}^{{-}{z}}{+}{{ⅇ}}^{{z}}}{{{ⅇ}}^{{-}{z}}{-}{{ⅇ}}^{{z}}}{,}{\mathrm{coth}}{}\left({z}\right){=}{I}{}\left({-}{I}{}{\mathrm{csch}}{}\left({2}{}{z}\right){-}{I}{}{\mathrm{coth}}{}\left({2}{}{z}\right)\right)\right]$ (4)
 > $\mathrm{eq1}≔\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{BesselI}\right)$
 ${\mathrm{eq1}}{≔}\left[{\mathrm{BesselI}}{}\left({a}{,}{-}{z}\right){=}\frac{{\left({-}{z}\right)}^{{a}}{}{\mathrm{BesselI}}{}\left({a}{,}{z}\right)}{{{z}}^{{a}}}{,}{\mathrm{BesselI}}{}\left({a}{,}{I}{}{z}\right){=}\frac{{\left({I}{}{z}\right)}^{{a}}{}{\mathrm{BesselJ}}{}\left({a}{,}{z}\right)}{{{z}}^{{a}}}{,}{\mathrm{BesselI}}{}\left({a}{+}{1}{,}{z}\right){}{\mathrm{BesselI}}{}\left({-}{a}{,}{z}\right){-}{\mathrm{BesselI}}{}\left({-}{1}{-}{a}{,}{z}\right){}{\mathrm{BesselI}}{}\left({a}{,}{z}\right){=}\frac{{2}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{a}\right)}{{\mathrm{π}}{}{z}}{,}{\mathrm{BesselI}}{}\left({a}{,}\sqrt{{{z}}^{{2}}}\right){=}\frac{{\left({{z}}^{{2}}\right)}^{\frac{{1}}{{2}}{}{a}}{}{\mathrm{BesselI}}{}\left({a}{,}{z}\right)}{{{z}}^{{a}}}{,}\left[{\mathrm{BesselI}}{}\left({a}{,}{b}{}{\left({c}{}{{z}}^{{q}}\right)}^{{p}}\right){=}\frac{{\left({b}{}{\left({c}{}{{z}}^{{q}}\right)}^{{p}}\right)}^{{a}}{}{\mathrm{BesselI}}{}\left({a}{,}{b}{}{{c}}^{{p}}{}{{z}}^{{p}{}{q}}\right)}{{\left({b}{}{{c}}^{{p}}{}{{z}}^{{p}{}{q}}\right)}^{{a}}}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{2}{}{p}{::}{\mathrm{integer}}\right]{,}{\mathrm{BesselI}}{}\left({a}{,}{z}\right){=}{-}\frac{{2}{}\left({a}{-}{1}\right){}{\mathrm{BesselI}}{}\left({a}{-}{1}{,}{z}\right)}{{z}}{+}{\mathrm{BesselI}}{}\left({a}{-}{2}{,}{z}\right){,}{\mathrm{BesselI}}{}\left({a}{,}{z}\right){=}\frac{{2}{}\left({a}{+}{1}\right){}{\mathrm{BesselI}}{}\left({a}{+}{1}{,}{z}\right)}{{z}}{+}{\mathrm{BesselI}}{}\left({a}{+}{2}{,}{z}\right)\right]$ (5)

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.

 > $\mathrm{depends}\left(\left[\mathrm{eq1}\right],a\right),\mathrm{depends}\left(\left[\mathrm{eq1}\right],z\right)$
 ${\mathrm{false}}{,}{\mathrm{false}}$ (6)

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

 > $\mathrm{eq2}≔\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{Ei}\left(a,z\right)\right)$
 ${\mathrm{eq2}}{≔}\left[{\mathrm{Ei}}{}\left({1}{,}{z}\right){=}{-}{\mathrm{Ei}}{}\left({-}{z}\right){+}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({-}{z}\right){-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({-}\frac{{1}}{{z}}\right){-}{\mathrm{ln}}{}\left({z}\right){,}\left[{\mathrm{Ei}}{}\left({-}{a}{,}{z}\right){=}{a}{!}{}{{ⅇ}}^{{-}{z}}{}\left({\sum }_{{\mathrm{_k1}}{=}{0}}^{{a}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{{z}}^{{\mathrm{_k1}}{-}{a}{-}{1}}}{{\mathrm{_k1}}{!}}\right){,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{::}{\mathrm{nonnegint}}\right]{,}{\mathrm{Ei}}{}\left({a}{,}{z}\right){=}\frac{{z}{}{\mathrm{Ei}}{}\left({-}{2}{+}{a}{,}{z}\right){+}\left({-}{2}{+}{a}{-}{z}\right){}{\mathrm{Ei}}{}\left({-}{1}{+}{a}{,}{z}\right)}{{-}{1}{+}{a}}{,}{\mathrm{Ei}}{}\left({a}{,}{z}\right){=}\frac{\left({-}{a}{+}{z}\right){}{\mathrm{Ei}}{}\left({1}{+}{a}{,}{z}\right){+}\left({1}{+}{a}\right){}{\mathrm{Ei}}{}\left({2}{+}{a}{,}{z}\right)}{{z}}\right]$ (7)
 > $\mathrm{depends}\left(\left[\mathrm{eq2}\right],a\right),\mathrm{depends}\left(\left[\mathrm{eq2}\right],z\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (8)
 >