BesselI - Maple Help

BesselI, BesselJ

The Bessel functions of the first kind

BesselK, BesselY

The Bessel functions of the second kind

HankelH1, HankelH2

The Hankel functions (Bessel functions of the third kind)

 Calling Sequence BesselI(v, x) BesselJ(v, x) BesselK(v, x) BesselY(v, x) HankelH1(v, x) HankelH2(v, x)

Parameters

 v - algebraic expression (the order or index) x - algebraic expression (the argument)

Description

 • BesselJ and BesselY are the Bessel functions of the first and second kinds, respectively.  They satisfy Bessel's equation:

${x}^{2}\mathrm{y\text{'}\text{'}}+x\mathrm{y\text{'}}+\left(-{v}^{2}+{x}^{2}\right)y=0$

 • BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively.  They satisfy the modified Bessel equation:

${x}^{2}\mathrm{y\text{'}\text{'}}+x\mathrm{y\text{'}}-\left({v}^{2}+{x}^{2}\right)y=0$

 • HankelH1 and HankelH2 are the Hankel functions, also known as the Bessel functions of the third kind.  They also satisfy Bessel's equation, and are related to BesselJ and BesselY by

$\mathrm{HankelH1}\left(v,x\right)=\mathrm{BesselJ}\left(v,x\right)+I\mathrm{BesselY}\left(v,x\right)$

$\mathrm{HankelH2}\left(v,x\right)=\mathrm{BesselJ}\left(v,x\right)-I\mathrm{BesselY}\left(v,x\right)$

Examples

 > $\mathrm{BesselJ}\left(0,2\right)$
 ${\mathrm{BesselJ}}{}\left({0}{,}{2}\right)$ (1)
 > $\mathrm{evalf}\left(\right)$
 ${0.2238907791}$ (2)
 > $\mathrm{BesselK}\left(1,-3.\right)$
 ${-0.04015643113}{-}{12.41987883}{}{I}$ (3)
 > $\mathrm{BesselI}\left(0,0\right)$
 ${1}$ (4)
 > $\mathrm{BesselY}\left(1.5+I,3.5-I\right)$
 ${0.9566518512}{-}{1.465483431}{}{I}$ (5)
 > $\mathrm{series}\left(\mathrm{BesselJ}\left(3,x\right),x\right)$
 $\frac{{1}}{{48}}{}{{x}}^{{3}}{-}\frac{{1}}{{768}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{7}}\right)$ (6)
 > $\mathrm{diff}\left(\mathrm{BesselJ}\left(v,x\right),x\right)$
 ${-}{\mathrm{BesselJ}}{}\left({v}{+}{1}{,}{x}\right){+}\frac{{v}{}{\mathrm{BesselJ}}{}\left({v}{,}{x}\right)}{{x}}$ (7)
 > $\mathrm{HankelH1}\left(2.5,3.7+I\right)$
 ${0.1809260572}{-}{0.08706107529}{}{I}$ (8)
 > $\mathrm{diff}\left(\mathrm{HankelH2}\left(v,{x}^{2}\right),x\right)$
 ${2}{}\left({-}{\mathrm{HankelH2}}{}\left({v}{+}{1}{,}{{x}}^{{2}}\right){+}\frac{{v}{}{\mathrm{HankelH2}}{}\left({v}{,}{{x}}^{{2}}\right)}{{{x}}^{{2}}}\right){}{x}$ (9)
 > $\mathrm{convert}\left(\mathrm{HankelH2}\left(v,x\right),\mathrm{Bessel}\right)$
 ${\mathrm{BesselJ}}{}\left({v}{,}{x}\right){-}{I}{}{\mathrm{BesselY}}{}\left({v}{,}{x}\right)$ (10)
 > $\mathrm{convert}\left(\mathrm{AiryAi}\left(x\right),\mathrm{Bessel}\right)$
 ${-}\frac{{x}{}{\mathrm{BesselI}}{}\left(\frac{{1}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}}}{{3}}\right)}{{3}{}{\left({{x}}^{{3}}\right)}^{{1}}{{6}}}}{+}\frac{{\left({{x}}^{{3}}\right)}^{{1}}{{6}}}{}{\mathrm{BesselI}}{}\left({-}\frac{{1}}{{3}}{,}\frac{{2}{}\sqrt{{{x}}^{{3}}}}{{3}}\right)}{{3}}$ (11)
 > $\mathrm{convert}\left(\mathrm{KelvinKer}\left(v,x\right),\mathrm{BesselK}\right)$
 $\frac{{\mathrm{BesselK}}{}\left({v}{,}\left(\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right){+}{\left({{ⅇ}}^{\frac{{I}}{{2}}{}{v}{}{\mathrm{\pi }}}\right)}^{{2}}{}{\mathrm{BesselK}}{}\left({v}{,}\left(\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)}{{2}{}{{ⅇ}}^{\frac{{I}}{{2}}{}{v}{}{\mathrm{\pi }}}}$ (12)