Airy - Maple Help

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convert/Airy

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

 Calling Sequence convert(expr, Airy)

Parameters

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

Description

 • convert/Airy converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Airy functions. The Airy functions are
 > FunctionAdvisor( Airy );
 The 2 functions in the "Airy" class are:
 $\left[{\mathrm{AiryAi}}{,}{\mathrm{AiryBi}}\right]$ (1)

Examples

 > $\mathrm{BesselK}\left(\frac{1}{3},z\right)$
 ${\mathrm{BesselK}}{}\left(\frac{{1}}{{3}}{,}{z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Airy}\right)$
 ${\mathrm{\pi }}{}\sqrt{\frac{{{3}}^{{1}}{{3}}}{}{{2}}^{{2}}{{3}}}}{{{z}}^{{2}}{{3}}}}}{}{\mathrm{AiryAi}}{}\left(\frac{{{3}}^{{2}}{{3}}}{}{{2}}^{{1}}{{3}}}{}{{z}}^{{2}}{{3}}}}{{2}}\right)$ (3)
 > $\mathrm{BesselJ}\left(\frac{1}{3},z\right)$
 ${\mathrm{BesselJ}}{}\left(\frac{{1}}{{3}}{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Airy}\right)$
 $\frac{\sqrt{\frac{{{3}}^{{1}}{{3}}}{}{{2}}^{{2}}{{3}}}}{{{z}}^{{2}}{{3}}}}}{}\left(\sqrt{{3}}{}{\mathrm{AiryAi}}{}\left({-}\frac{{{3}}^{{2}}{{3}}}{}{{2}}^{{1}}{{3}}}{}{{z}}^{{2}}{{3}}}}{{2}}\right){-}{\mathrm{AiryBi}}{}\left({-}\frac{{{3}}^{{2}}{{3}}}{}{{2}}^{{1}}{{3}}}{}{{z}}^{{2}}{{3}}}}{{2}}\right)\right)}{{2}}$ (5)
 > $\mathrm{BesselI}\left(-\frac{1}{3},\frac{2}{3}{z}^{\frac{3}{2}}\right)$
 ${\mathrm{BesselI}}{}\left({-}\frac{{1}}{{3}}{,}\frac{{2}{}{{z}}^{{3}}{{2}}}}{{3}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{Airy}\right)$
 $\frac{\sqrt{{3}}{}\left(\sqrt{{3}}{}{\mathrm{AiryAi}}{}\left({\left({{z}}^{{3}}{{2}}}\right)}^{{2}}{{3}}}\right){+}{\mathrm{AiryBi}}{}\left({\left({{z}}^{{3}}{{2}}}\right)}^{{2}}{{3}}}\right)\right)}{{2}{}{\left({{z}}^{{3}}{{2}}}\right)}^{{1}}{{3}}}}$ (7)
 > $\mathrm{KummerM}\left(\frac{5}{6},\frac{5}{3},2Iz\right)$
 ${\mathrm{KummerM}}{}\left(\frac{{5}}{{6}}{,}\frac{{5}}{{3}}{,}{2}{}{I}{}{z}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{Airy}\right)$
 $\frac{{{ⅇ}}^{{I}{}{z}}{}{\mathrm{\pi }}{}\sqrt{{3}}{}{{2}}^{{1}}{{3}}}{}\sqrt{\frac{{{3}}^{{1}}{{3}}}{}{{2}}^{{2}}{{3}}}}{{{z}}^{{2}}{{3}}}}}{}\left(\sqrt{{3}}{}{\mathrm{AiryAi}}{}\left({-}\frac{{{3}}^{{2}}{{3}}}{}{{2}}^{{1}}{{3}}}{}{{z}}^{{2}}{{3}}}}{{2}}\right){-}{\mathrm{AiryBi}}{}\left({-}\frac{{{3}}^{{2}}{{3}}}{}{{2}}^{{1}}{{3}}}{}{{z}}^{{2}}{{3}}}}{{2}}\right)\right)}{{9}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{{z}}^{{1}}{{3}}}}$ (9)
 > $\mathrm{MeijerG}\left(\left[\left[\right],\left[\frac{1}{6},\frac{2}{3}\right]\right],\left[\left[\frac{1}{3},0\right],\left[\frac{2}{3},\frac{1}{6}\right]\right],z\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\frac{{1}}{{6}}{,}\frac{{2}}{{3}}\right]\right]{,}\left[\left[\frac{{1}}{{3}}{,}{0}\right]{,}\left[\frac{{2}}{{3}}{,}\frac{{1}}{{6}}\right]\right]{,}{z}\right)$ (10)
 > $\mathrm{convert}\left(,\mathrm{Airy}\right)$
 $\frac{\left(\sqrt{{3}}{}\left({{z}}^{{1}}{{3}}}{+}{\left({-}{z}\right)}^{{1}}{{3}}}\right){}{\mathrm{AiryAi}}{}\left({-}{{3}}^{{2}}{{3}}}{}{\left({-}{z}\right)}^{{1}}{{3}}}\right){+}\left({-}{{z}}^{{1}}{{3}}}{+}{\left({-}{z}\right)}^{{1}}{{3}}}\right){}{\mathrm{AiryBi}}{}\left({-}{{3}}^{{2}}{{3}}}{}{\left({-}{z}\right)}^{{1}}{{3}}}\right)\right){}\sqrt{\frac{{{3}}^{{1}}{{3}}}}{{\left({-}{z}\right)}^{{1}}{{3}}}}}}{{4}{}{\left({-}{z}\right)}^{{1}}{{6}}}{}{\mathrm{\pi }}}$ (11)

 See Also