specialize a given mathematical function into other mathematical functions

Parameters

 specialize - literal name; 'specialize' math_function_1 - (Maple) name of mathematical function to be specialized math_function_1(a, b, ..., z) - (Maple) mathematical function to be specialized, evaluated at (a, b, ..., z) math_function_2 - (optional) name of mathematical function used to express output

Description

 • The FunctionAdvisor(specialize, math_function_1) command attempts to specialize this function in terms of every other mathematical function. When the function is given with parameters, as in math_function_1(a, b, ..., z), only specializations valid for those values of the parameters or particular cases of them are returned.
 • The FunctionAdvisor(specialize, math_function_1, math_function_2) command attempts to specialize the first function, math_function_1 in terms of the other one using Maple algorithms. When the first function is given with parameters, as in math_function_1(a, b, ..., z), only specializations valid for those values of the parameters or particular cases of them are returned. For more information, see convert/to_special_function.
 • Note the specialization operation typically requires additional constraints for the function parameters involved. These constraints are returned as a boolean function. If no constraints are required, this is explicitly indicated.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{sin},\mathrm{hypergeom}\right)$
 $\left[{\mathrm{sin}}{}\left({z}\right){=}{z}{}{\mathrm{hypergeom}}{}\left(\left[\right]{,}\left[\frac{{3}}{{2}}\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{hypergeom},\mathrm{sin}\right)$
 $\left[{\mathrm{hypergeom}}{}\left(\left[\right]{,}\left[{a}\right]{,}{z}\right){=}\frac{{\mathrm{sin}}{}\left({2}{}\sqrt{{-}{z}}\right)}{{2}{}\sqrt{{-}{z}}}{,}{a}{=}\frac{{3}}{{2}}\right]$ (2)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{hypergeom},\mathrm{dilog}\right)$
 $\left[{\mathrm{hypergeom}}{}\left(\left[{a}{,}{b}{,}{c}\right]{,}\left[{d}{,}{e}\right]{,}{z}\right){=}\frac{{\mathrm{dilog}}{}\left({1}{-}{z}\right)}{{z}}{,}{a}{=}{1}{\wedge }{b}{=}{1}{\wedge }{c}{=}{1}{\wedge }{d}{=}{2}{\wedge }{e}{=}{2}\right]$ (3)

The specialization can be requested for particular values of some or all of the function parameters.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{hypergeom}\left(\left[1,1,a\right],\left[b,c\right],1\right),\mathrm{dilog}\right)$
 $\left[{\mathrm{hypergeom}}{}\left(\left[{1}{,}{1}{,}{a}\right]{,}\left[{b}{,}{c}\right]{,}{1}\right){=}{\mathrm{dilog}}{}\left({0}\right){,}{a}{=}{1}{\wedge }{b}{=}{2}{\wedge }{c}{=}{2}\right]$ (4)

Note the difference between using specialize and relate (see FunctionAdvisor/relate). The former computes the required restrictions on the function parameters while the latter only returns a result when it does not require additional constraints on the parameters involved.

The following is a specialization which cannot be obtained using the relate keyword.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{HermiteH},\mathrm{KummerU}\right)$
 $\left[{\mathrm{HermiteH}}{}\left({a}{,}{z}\right){=}{{2}}^{{a}}{}{\mathrm{KummerU}}{}\left({-}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{,}{{z}}^{{2}}\right){,}{0}{<}{\mathrm{\Re }}{}\left({z}\right){\vee }\left({\mathrm{\Re }}{}\left({z}\right){=}{0}{\wedge }{0}{<}{\mathrm{\Im }}{}\left({z}\right)\right)\right]$ (5)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{KummerU},\mathrm{HermiteH}\right)$
 $\left[{\mathrm{KummerU}}{}\left({a}{,}{b}{,}{z}\right){=}{\mathrm{HermiteH}}{}\left({-}{2}{}{a}{,}\sqrt{{z}}\right){}{{2}}^{{2}{}{a}}{,}{b}{=}\frac{{1}}{{2}}\right]{,}\left[{\mathrm{KummerU}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{\mathrm{HermiteH}}{}\left({-}{2}{}{a}{+}{1}{,}\sqrt{{z}}\right){}{{2}}^{{2}{}{a}}}{{2}{}\sqrt{{z}}}{,}{b}{=}\frac{{3}}{{2}}\right]$ (6)

Knowing the possible specialization of a function is useful when trying to express functions in terms of other functions. Typically, the conversion is not possible unless restrictions on the function parameters are imposed. To perform the transformation (conversion), use the output of the FunctionAdvisor command with convert and assuming. For example, to transform as in the previous example, all KummerU functions entering an expression try the following.

 > $\mathrm{convert}\left(2\mathrm{sin}\left(z\right)+\mathrm{KummerU}\left(a,b,z\right),\mathrm{HermiteH}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}b=\frac{3}{2}$
 ${2}{}{\mathrm{sin}}{}\left({z}\right){+}\frac{{\mathrm{HermiteH}}{}\left({-}{2}{}{a}{+}{1}{,}\sqrt{{z}}\right){}{{2}}^{{2}{}{a}}}{{2}{}\sqrt{{z}}}$ (7)

The following example specializes sin into the other mathematical functions.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{sin}\right)$
 $\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{\sqrt{{z}}{}\sqrt{{\mathrm{\pi }}}{}\sqrt{{2}}{}{\mathrm{BesselJ}}{}\left(\frac{{1}}{{2}}{,}{z}\right)}{{2}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{\mathrm{ChebyshevT}}{}\left({b}{,}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}{-}{2}{}{z}}{{2}{}{b}}\right)\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{\mathrm{ChebyshevU}}{}\left({-}\frac{{-}{z}{+}{\mathrm{arccos}}{}\left({c}\right)}{{\mathrm{arccos}}{}\left({c}\right)}{,}{c}\right){}\sqrt{{-}{{c}}^{{2}}{+}{1}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{I}{-}{2}{}{z}{}{{\mathrm{Ei}}}_{{0}}{}\left({-}{2}{}{I}{}{z}\right)}{{2}{}{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{-}\frac{{I}}{{2}}{}\left({\mathrm{\Gamma }}{}\left({1}{,}{-}{2}{}{I}{}{z}\right){-}{1}\right)}{{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{HeunB}}{}\left({2}{,}{0}{,}{0}{,}{0}{,}\sqrt{{2}}{}\sqrt{{I}{}{z}}\right)}{{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{\left({2}{}{I}{}{{z}}^{{2}}{+}{z}\right){}{\mathrm{HeunC}}{}\left({1}{,}{1}{,}{1}{,}{0}{,}\frac{{1}}{{2}}{,}{-}{2}{}{I}{}{z}\right)}{{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{z}{}{\mathrm{KummerM}}{}\left({1}{,}{2}{,}{2}{}{I}{}{z}\right)}{{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{-}\frac{{I}}{{2}}{}\left({-}{1}{+}{{ⅇ}}^{{2}{}{I}{}{z}}\right)}{{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\sqrt{{\mathrm{\pi }}}{}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[\frac{{1}}{{2}}\right]{,}\left[{0}\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{\sqrt{{z}}{}{\mathrm{StruveH}}{}\left({-}\frac{{1}}{{2}}{,}{z}\right){}\sqrt{{\mathrm{\pi }}}{}\sqrt{{2}}}{{2}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{I}}{{2}}{}{\mathrm{WhittakerM}}{}\left({0}{,}\frac{{1}}{{2}}{,}{2}{}{I}{}{z}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-}{\mathrm{cos}}{}\left({z}{+}\frac{{\mathrm{\pi }}}{{2}}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-}{\mathrm{cosh}}{}\left({I}{}\left({z}{+}\frac{{\mathrm{\pi }}}{{2}}\right)\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{2}{}{\mathrm{cot}}{}\left(\frac{{z}}{{2}}\right)}{{{\mathrm{cot}}{}\left(\frac{{z}}{{2}}\right)}^{{2}}{+}{1}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{-}{2}{}{I}{}{\mathrm{coth}}{}\left(\frac{{I}}{{2}}{}{z}\right)}{{{\mathrm{coth}}{}\left(\frac{{I}}{{2}}{}{z}\right)}^{{2}}{-}{1}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{1}}{{\mathrm{csc}}{}\left({z}\right)}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{-I}}{{\mathrm{csch}}{}\left({I}{}{z}\right)}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{I}}{{2}}{}\left({{ⅇ}}^{{I}{}{z}}{-}\frac{{1}}{{{ⅇ}}^{{I}{}{z}}}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{z}{}{\mathrm{hypergeom}}{}\left(\left[\right]{,}\left[\frac{{3}}{{2}}\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{1}}{{\mathrm{sec}}{}\left({z}{+}\frac{{\mathrm{\pi }}}{{2}}\right)}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{1}}{{\mathrm{sech}}{}\left({I}{}\left({z}{+}\frac{{\mathrm{\pi }}}{{2}}\right)\right)}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}{-I}{}{\mathrm{sinh}}{}\left({I}{}{z}\right){,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{2}{}{\mathrm{tan}}{}\left(\frac{{z}}{{2}}\right)}{{1}{+}{{\mathrm{tan}}{}\left(\frac{{z}}{{2}}\right)}^{{2}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]{,}\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{2}{}{I}{}{\mathrm{tanh}}{}\left(\frac{{I}}{{2}}{}{z}\right)}{{{\mathrm{tanh}}{}\left(\frac{{I}}{{2}}{}{z}\right)}^{{2}}{-}{1}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]$ (8)

While the previous specializations do not require restrictions on the function parameters, if you specialize more general functions into simpler ones, restrictions are necessary. For example, for the LegendreQ function you have the following.

 > $\left[\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{LegendreQ}\right)\right]$
 $\left[\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{AppellF1}}{}\left({1}{+}\frac{{a}}{{2}}{,}{0}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{AppellF1}}{}\left({1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}{0}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{AppellF2}}{}\left({1}{+}\frac{{a}}{{2}}{,}{0}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}{1}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{AppellF2}}{}\left({1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}{0}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{,}{1}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{AppellF3}}{}\left({0}{,}{1}{+}\frac{{a}}{{2}}{,}{0}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{AppellF3}}{}\left({0}{,}{1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}{0}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{AppellF4}}{}\left({1}{+}\frac{{a}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}{1}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{AppellF4}}{}\left({1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{,}{1}{,}\frac{{3}}{{2}}{+}{a}{,}{0}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{{{z}}^{{2}}{}{\left(\frac{{{z}}^{{2}}{-}{1}}{{{z}}^{{2}}}\right)}^{{3}}{{4}}}{}\sqrt{{2}}{}{\mathrm{EllipticK}}{}\left(\frac{\sqrt{{2}}{}\sqrt{\frac{{{z}}^{{2}}{+}\sqrt{{{z}}^{{4}}{-}{{z}}^{{2}}}{-}{1}}{{{z}}^{{2}}{-}{1}}}}{{2}}\right)}{\left({{z}}^{{2}}{-}{1}\right){}\sqrt{\frac{\sqrt{{{z}}^{{4}}{-}{{z}}^{{2}}}}{\sqrt{\frac{{-}{{z}}^{{2}}{+}\sqrt{{{z}}^{{4}}{-}{{z}}^{{2}}}{+}{1}}{{{z}}^{{2}}{-}{1}}}{}\left({{z}}^{{2}}{-}{1}\right){}\sqrt{\frac{{{z}}^{{2}}{+}\sqrt{{{z}}^{{4}}{-}{{z}}^{{2}}}{-}{1}}{{{z}}^{{2}}{-}{1}}}}}}{,}{a}{=}{-}\frac{{1}}{{2}}\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{HeunC}}{}\left({0}{,}\frac{{1}}{{2}}{+}{a}{,}{-}\frac{{1}}{{2}}{,}{0}{,}\frac{{1}}{{4}}{}{a}{+}\frac{{3}}{{8}}{+}\frac{{1}}{{4}}{}{{a}}^{{2}}{,}\frac{{1}}{{{z}}^{{2}}{}\left(\frac{{1}}{{{z}}^{{2}}}{-}{1}\right)}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{\left({1}{-}\frac{{1}}{{{z}}^{{2}}}\right)}^{\frac{{a}}{{2}}{+}\frac{{1}}{{2}}}{}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{HeunC}}{}\left({0}{,}\frac{{1}}{{2}}{+}{a}{,}{-}\frac{{1}}{{2}}{,}{0}{,}\frac{{1}}{{4}}{}{a}{+}\frac{{3}}{{8}}{-}\frac{{1}}{{4}}{}{{b}}^{{2}}{+}\frac{{1}}{{4}}{}{{a}}^{{2}}{,}\frac{{1}}{{{z}}^{{2}}{}\left(\frac{{1}}{{{z}}^{{2}}}{-}{1}\right)}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{\left({1}{-}\frac{{1}}{{{z}}^{{2}}}\right)}^{\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}}{}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{HeunG}}{}\left({0}{,}{0}{,}{1}{+}\frac{{a}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}{0}{,}{1}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{HeunG}}{}\left({0}{,}{0}{,}{1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{,}{0}{,}{1}{+}{b}{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{{\mathrm{\pi }}{}{\mathrm{JacobiP}}{}\left({-}{1}{-}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}{a}{,}{0}{,}\frac{{{z}}^{{2}}{-}{2}}{{{z}}^{{2}}}\right){}{{2}}^{{1}{+}{a}}}{{4}{}{{z}}^{{1}{+}{a}}{}{{2}}^{{a}}{}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}{}\left({2}{+}{a}\right)}{{2}}\right)}{,}{\mathrm{with no restrictions on}}{}\left({a}{,}{z}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({1}{+}{b}{+}{a}\right){}{\mathrm{\Gamma }}{}\left({-}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}\right){}{\mathrm{JacobiP}}{}\left({-}{1}{-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{,}\frac{{1}}{{2}}{+}{a}{,}{b}{,}\frac{{{z}}^{{2}}{-}{2}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{1}{+}{b}{+}{a}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}\right){}{{2}}^{{a}}}{,}{\mathrm{with no restrictions on}}{}\left({a}{,}{b}{,}{z}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{\mathrm{\pi }}{}{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}{\mathrm{csc}}{}\left({b}{}{\mathrm{\pi }}\right){}\left({\mathrm{LegendreP}}{}\left({a}{,}{b}{,}{z}\right){-}\frac{{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{LegendreP}}{}\left({a}{,}{-}{b}{,}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}{-}{b}{+}{1}\right)}\right)}{{2}}{,}{b}{::}\left({¬}{'}{\mathrm{integer}}{'}\right){\wedge }\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonnegint}}{'}\right){\wedge }\left({a}{-}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonnegint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{{\mathrm{MeijerG}}{}\left(\left[\left[{-}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{-}\frac{{a}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}\frac{{1}}{{2}}{-}{a}\right]\right]{,}{-}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{1}{+}{a}}}{,}{\mathrm{with no restrictions on}}{}\left({a}{,}{z}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{MeijerG}}{}\left(\left[\left[{-}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{-}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}\frac{{1}}{{2}}{-}{a}\right]\right]{,}{-}\frac{{1}}{{{z}}^{{2}}}\right){}{{2}}^{{b}}}{{2}{}{{z}}^{{1}{+}{b}{+}{a}}}{,}{\mathrm{with no restrictions on}}{}\left({a}{,}{b}{,}{z}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{{I}}{{2}}{}\left({-}{\mathrm{\pi }}{+}{2}{}{\mathrm{arccot}}{}\left(\frac{{I}}{{z}}\right)\right){,}{a}{=}{0}\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{{\mathrm{\pi }}{}\sqrt{{-}\frac{{\left({-}{1}{+}{z}\right)}^{{2}}}{{{z}}^{{2}}}}{}{z}{+}{2}{}{\mathrm{arccoth}}{}\left(\frac{{1}}{{z}}\right){}{z}{-}{2}{}{\mathrm{arccoth}}{}\left(\frac{{1}}{{z}}\right)}{{2}{}\left({-}{1}{+}{z}\right)}{,}{a}{=}{0}\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}{\mathrm{arctanh}}{}\left(\frac{{1}}{{z}}\right){,}{a}{=}{0}\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{z}\right){=}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{hypergeom}}{}\left(\left[{1}{+}\frac{{a}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}\right]{,}\left[\frac{{3}}{{2}}{+}{a}\right]{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{ⅇ}}^{{I}{}{b}{}{\mathrm{\pi }}}{}\sqrt{{\mathrm{\pi }}}{}{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right){}{\mathrm{hypergeom}}{}\left(\left[{1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}\right]{,}\left[\frac{{3}}{{2}}{+}{a}\right]{,}\frac{{1}}{{{z}}^{{2}}}\right)}{{2}{}{{z}}^{{a}{+}{b}{+}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{2}}{+}{a}\right){}{{2}}^{{a}}}{,}\left({a}{+}{b}{+}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left(\frac{{3}}{{2}}{+}{a}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]\right]$ (9)
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