Kelvin - Maple Help

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

 > ${\mathrm{PDEtools}}_{\mathrm{dpolyform}}\left(\mathrm{sol},\mathrm{no_Fn}\right)$
 $\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right){=}\frac{\left({-}{2}{}{a}{+}{z}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right)\right)}{{z}}{+}\frac{\left({1}{+}{2}{}{a}\right){}{y}{}\left({z}\right)}{{2}{}{z}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\right]$ (5)

sol can be rewritten using Kelvin functions

 > $\mathrm{convert}\left(\mathrm{sol},\mathrm{Kelvin}\right)$
 ${y}{}\left({z}\right){=}\frac{\mathrm{c__1}{}{{ⅇ}}^{\frac{{z}}{{2}}}{}{\mathrm{\Gamma }}{}\left({1}{+}{a}\right){}\left({\mathrm{KelvinBer}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right){+}{I}{}{\mathrm{KelvinBei}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right)\right){}{{2}}^{{a}}}{{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}}{+}\frac{\mathrm{c__2}{}\left(\frac{{{2}}^{{a}}{}{\mathrm{KelvinBer}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right)}{{4}{}{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{-}{1}{+}{2}{}{a}}}{-}\frac{{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{1}{+}{2}{}{a}}{}{\mathrm{KelvinBer}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right)}{{4}{}{{z}}^{{2}{}{a}}{}{{2}}^{{a}}}{-}\frac{{I}{}{\mathrm{KelvinBei}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right){}{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{1}{+}{2}{}{a}}}{{4}{}{{z}}^{{2}{}{a}}{}{{2}}^{{a}}}{+}\frac{{I}{}{\mathrm{KelvinBei}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right){}{{2}}^{{a}}}{{4}{}{\left(\frac{{I}}{{2}}{}{z}\right)}^{{a}}{}{{2}}^{{-}{1}{+}{2}{}{a}}}\right){}\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{\frac{{z}}{{2}}}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}\left({1}{+}{a}\right)\right)}$ (6)
 > $\mathrm{collect}\left(,\left[\mathrm{KelvinBer},\mathrm{KelvinBei}\right],\mathrm{simplify}\right)$
 ${y}{}\left({z}\right){=}\frac{{{ⅇ}}^{\frac{{z}}{{2}}}{}\left({2}{}\mathrm{c__1}{}{\mathrm{\Gamma }}{}\left({1}{+}{a}\right){}{{4}}^{{a}}{-}\mathrm{c__2}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{a}\right)\right){}{\left({I}{}{z}\right)}^{{-}{a}}{}{\mathrm{KelvinBer}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right)}{{2}}{+}\frac{\mathrm{c__2}{}{{z}}^{{-}{2}{}{a}}{}\sqrt{{\mathrm{\pi }}}{}{\left({I}{}{z}\right)}^{{a}}{}{{ⅇ}}^{\frac{{z}}{{2}}}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{a}\right){}{\mathrm{KelvinBer}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right)}{{2}}{+}{I}{}\left(\mathrm{c__1}{}{\mathrm{\Gamma }}{}\left({1}{+}{a}\right){}{{4}}^{{a}}{-}\frac{\mathrm{c__2}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{a}\right)}{{2}}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}{}{\left({I}{}{z}\right)}^{{-}{a}}{}{\mathrm{KelvinBei}}{}\left({a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right){+}\frac{{I}{}\mathrm{c__2}{}{{z}}^{{-}{2}{}{a}}{}\sqrt{{\mathrm{\pi }}}{}{\left({I}{}{z}\right)}^{{a}}{}{{ⅇ}}^{\frac{{z}}{{2}}}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{a}\right){}{\mathrm{KelvinBei}}{}\left({-}{a}{,}\left(\frac{{1}}{{4}}{-}\frac{{I}}{{4}}\right){}{z}{}\sqrt{{2}}\right)}{{2}}$ (7)