convert/Kelvin - Maple Programming Help

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convert/Kelvin

convert special functions admitting 1F1 or 0F1 hypergeometric representation into Kelvin functions

 Calling Sequence convert(expr, Kelvin)

Parameters

 expr - Maple expression, equation, or a set or list of them

Description

 • convert/Kelvin converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Kelvin functions. The Kelvin functions are
 > FunctionAdvisor( Kelvin );
 The 6 functions in the "Kelvin" class are:
 $\left[{\mathrm{KelvinBei}}{,}{\mathrm{KelvinBer}}{,}{\mathrm{KelvinHei}}{,}{\mathrm{KelvinHer}}{,}{\mathrm{KelvinKei}}{,}{\mathrm{KelvinKer}}\right]$ (1)

Examples

 > $\mathrm{AiryAi}\left(z\right)$
 ${\mathrm{AiryAi}}{}\left({z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Kelvin}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0<\mathrm{ℜ}\left(z\right)$
 $\frac{{1}}{{6}}{}\frac{\sqrt{{3}}{}\sqrt{{z}}{}\left({\mathrm{KelvinKer}}{}\left(\frac{{1}}{{3}}{,}\left(\frac{{1}}{{3}}{-}\frac{{1}}{{3}}{}{I}\right){}{{z}}^{{3}{/}{2}}{}\sqrt{{2}}\right){+}{I}{}{\mathrm{KelvinKei}}{}\left(\frac{{1}}{{3}}{,}\left(\frac{{1}}{{3}}{-}\frac{{1}}{{3}}{}{I}\right){}{{z}}^{{3}{/}{2}}{}\sqrt{{2}}\right)\right){}\left(\sqrt{{3}}{+}{I}\right)}{{\mathrm{π}}}$ (3)

Consider the following expression

 > $\mathrm{sol}≔y\left(z\right)=\mathrm{_C1}\mathrm{KummerM}\left(\frac{1}{2}+a,1+2a,z\right)+\mathrm{_C2}\mathrm{KummerU}\left(\frac{1}{2}+a,1+2a,z\right)$
 ${\mathrm{sol}}{≔}{y}{}\left({z}\right){=}{\mathrm{_C1}}{}{\mathrm{KummerM}}{}\left(\frac{{1}}{{2}}{+}{a}{,}{1}{+}{2}{}{a}{,}{z}\right){+}{\mathrm{_C2}}{}{\mathrm{KummerU}}{}\left(\frac{{1}}{{2}}{+}{a}{,}{1}{+}{2}{}{a}{,}{z}\right)$ (4)

where sol is the solution to the following linear differential equation (see dpolyform)

 > $\mathrm{PDEtools}[\mathrm{dpolyform}]\left(\mathrm{sol},\mathrm{no_Fn}\right)$
 $\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}{}{y}{}\left({z}\right){=}\frac{\left({-}{2}{}{a}{+}{z}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{y}{}\left({z}\right)\right)}{{z}}{+}\frac{{1}}{{2}}{}\frac{\left({1}{+}{2}{}{a}\right){}{y}{}\left({z}\right)}{{z}}\right]{&where}\left[{}\right]$ (5)

sol can be rewritten using Kelvin functions

 > $\mathrm{convert}\left(\mathrm{sol},\mathrm{Kelvin}\right)$
 ${y}{}\left({z}\right){=}\frac{{\mathrm{_C1}}{}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}{\mathrm{Γ}}{}\left({1}{+}{a}\right){}\left({\mathrm{KelvinBer}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right){+}{I}{}{\mathrm{KelvinBei}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)\right){}{{2}}^{{a}}}{{\left(\frac{{1}}{{2}}{}{I}{}{z}\right)}^{{a}}}{+}\frac{{\mathrm{_C2}}{}\left(\frac{{1}}{{4}}{}\frac{{{2}}^{{a}}{}{\mathrm{KelvinBer}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)}{{\left(\frac{{1}}{{2}}{}{I}{}{z}\right)}^{{a}}{}{{2}}^{{-}{1}{+}{2}{}{a}}}{-}\frac{{1}}{{4}}{}\frac{{\left(\frac{{1}}{{2}}{}{I}{}{z}\right)}^{{a}}{}{{2}}^{{1}{+}{2}{}{a}}{}{\mathrm{KelvinBer}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)}{{{z}}^{{2}{}{a}}{}{{2}}^{{a}}}{-}\frac{\frac{{1}}{{4}}{}{I}{}{\mathrm{KelvinBei}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right){}{\left(\frac{{1}}{{2}}{}{I}{}{z}\right)}^{{a}}{}{{2}}^{{1}{+}{2}{}{a}}}{{{z}}^{{2}{}{a}}{}{{2}}^{{a}}}{+}\frac{\frac{{1}}{{4}}{}{I}{}{\mathrm{KelvinBei}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right){}{{2}}^{{a}}}{{\left(\frac{{1}}{{2}}{}{I}{}{z}\right)}^{{a}}{}{{2}}^{{-}{1}{+}{2}{}{a}}}\right){}\sqrt{{\mathrm{π}}}{}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}}{{\mathrm{sin}}{}\left({\mathrm{π}}{}\left({1}{+}{a}\right)\right)}$ (6)
 > $\mathrm{collect}\left(,\left[\mathrm{KelvinBer},\mathrm{KelvinBei}\right],\mathrm{simplify}\right)$
 ${y}{}\left({z}\right){=}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}\left({2}{}{\mathrm{_C1}}{}{\mathrm{Γ}}{}\left({a}\right){}{a}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{a}\right){}{{4}}^{{a}}{-}{\mathrm{_C2}}{}\sqrt{{\mathrm{π}}}\right){}{\left({I}{}{z}\right)}^{{-}{a}}{}{\mathrm{KelvinBer}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)}{{\mathrm{sin}}{}\left({\mathrm{π}}{}{a}\right)}{-}\frac{{1}}{{2}}{}\frac{{\mathrm{_C2}}{}{{z}}^{{-}{2}{}{a}}{}{\left({I}{}{z}\right)}^{{a}}{}\sqrt{{\mathrm{π}}}{}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}{\mathrm{KelvinBer}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)}{{\mathrm{sin}}{}\left({\mathrm{π}}{}\left({1}{+}{a}\right)\right)}{+}\frac{\frac{{1}}{{4}}{}{I}{}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}{\left({I}{}{z}\right)}^{{-}{a}}{}\left({{2}}^{{2}{+}{2}{}{a}}{}{\mathrm{_C1}}{}{\mathrm{Γ}}{}\left({a}\right){}{a}{}{\mathrm{sin}}{}\left({\mathrm{π}}{}{a}\right){-}{2}{}{\mathrm{_C2}}{}\sqrt{{\mathrm{π}}}\right){}{\mathrm{KelvinBei}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)}{{\mathrm{sin}}{}\left({\mathrm{π}}{}{a}\right)}{-}\frac{\frac{{1}}{{2}}{}{I}{}{\mathrm{_C2}}{}{{z}}^{{-}{2}{}{a}}{}{\left({I}{}{z}\right)}^{{a}}{}\sqrt{{\mathrm{π}}}{}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}{}{\mathrm{KelvinBei}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{1}}{{4}}{}{I}\right){}{z}{}\sqrt{{2}}\right)}{{\mathrm{sin}}{}\left({\mathrm{π}}{}\left({1}{+}{a}\right)\right)}$ (7)