Bessel - Maple Help

convert/Bessel

convert an expression with Airy wave functions or Bessel functions into Bessel functions

 Calling Sequence convert(expr, Bessel) convert(expr, BesselI) convert(expr, BesselJ) convert(expr, BesselK) convert(expr, BesselY) convert(expr, Hankel)

Parameters

 expr - Maple expression

Description

 • convert/Bessel converts Airy wave functions, their derivatives, and Bessel functions in an expression to the Bessel functions.
 • The option Bessel will give the simplest conversion. The other options will force a conversion to an expression in that function, if it is possible.

Examples

 > $\mathrm{convert}\left(\mathrm{AiryAi}\left(x\right),\mathrm{Bessel}\right)$
 ${-}\frac{{x}{}{\mathrm{BesselI}}{}\left(\frac{{1}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}}}{{3}}\right)}{{3}{}{\left({{x}}^{{3}}\right)}^{{1}}{{6}}}}{+}\frac{{\left({{x}}^{{3}}\right)}^{{1}}{{6}}}{}{\mathrm{BesselI}}{}\left({-}\frac{{1}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}}}{{3}}\right)}{{3}}$ (1)
 > $\mathrm{convert}\left(-\mathrm{AiryBi}\left(\mathrm{sqrt}\left(x\right)\right),\mathrm{Bessel}\right)$
 ${-}\frac{\sqrt{{3}}{}\left(\sqrt{{x}}{}{\mathrm{BesselI}}{}\left(\frac{{1}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}{{2}}}}}{{3}}\right){+}{\mathrm{BesselI}}{}\left({-}\frac{{1}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}{{2}}}}}{{3}}\right){}{\left({{x}}^{{3}}{{2}}}\right)}^{{1}}{{3}}}\right)}{{3}{}{\left({{x}}^{{3}}{{2}}}\right)}^{{1}}{{6}}}}$ (2)
 > $\mathrm{convert}\left(\mathrm{AiryAi}\left(1,x\right),\mathrm{BesselI}\right)$
 $\frac{{{x}}^{{2}}{}{\mathrm{BesselI}}{}\left(\frac{{2}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}}}{{3}}\right)}{{3}{}{\left({{x}}^{{3}}\right)}^{{1}}{{3}}}}{-}\frac{{\left({{x}}^{{3}}\right)}^{{1}}{{3}}}{}{\mathrm{BesselI}}{}\left({-}\frac{{2}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}}}{{3}}\right)}{{3}}$ (3)
 > $\mathrm{convert}\left(\mathrm{sin}\left(\mathrm{HankelH1}\left(v,x\right)\right),\mathrm{Bessel}\right)$
 ${\mathrm{sin}}{}\left({\mathrm{BesselJ}}{}\left({v}{,}{x}\right){+}{I}{}{\mathrm{BesselY}}{}\left({v}{,}{x}\right)\right)$ (4)
 > $\mathrm{convert}\left(\mathrm{BesselJ}\left(v,2\right),\mathrm{BesselY}\right)$
 $\frac{{\mathrm{BesselY}}{}\left({-}{v}{,}{2}\right){-}{\mathrm{BesselY}}{}\left({v}{,}{2}\right){}{\mathrm{cos}}{}\left({v}{}{\mathrm{\pi }}\right)}{{\mathrm{sin}}{}\left({v}{}{\mathrm{\pi }}\right)}$ (5)
 > $\mathrm{convert}\left(\mathrm{KelvinKei}\left(v,{z}^{2}\right),\mathrm{BesselK}\right)$
 $\frac{\frac{{I}}{{2}}{}\left({\left({{ⅇ}}^{\frac{{I}}{{2}}{}{v}{}{\mathrm{\pi }}}\right)}^{{2}}{}{\mathrm{BesselK}}{}\left({v}{,}\left(\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{{z}}^{{2}}{}\sqrt{{2}}\right){-}{\mathrm{BesselK}}{}\left({v}{,}\left(\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{{z}}^{{2}}{}\sqrt{{2}}\right)\right)}{{{ⅇ}}^{\frac{{I}}{{2}}{}{v}{}{\mathrm{\pi }}}}$ (6)