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MTM[besseli], MTM[besselj]

Bessel functions of the first kind

MTM[besselk], MTM[bessely]

Bessel functions of the second kind

MTM[besselh]

Bessel functions of the third kind

 Calling Sequence besseli(v,x) besselj(v,x) besselk(v,x) bessely(v,x) besselh(v,x) besselh(v,k,x) besselh[k](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 the Bessel equation:

${x}^{2}\frac{{{\partial }}^{2}}{{\partial }{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y+x\frac{{\partial }}{{\partial }x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y+\left(-{v}^{2}+{x}^{2}\right)y=0$

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

${x}^{2}\frac{{{\partial }}^{2}}{{\partial }{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y+x\frac{{\partial }}{{\partial }x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y-\left(-{v}^{2}+{x}^{2}\right)y=0$

 • besselh are the Bessel functions of the third kind, also known as Hankel functions. They are linear combinations of the preceding Bessel functions:

$\mathrm{besselh}\left(v,1,x\right)=\mathrm{besselj}\left(v,x\right)+i\mathrm{bessely}\left(v,x\right)$

$\mathrm{besselh}\left(v,2,x\right)=\mathrm{besselj}\left(v,x\right)-i\mathrm{bessely}\left(v,x\right)$

 Note that besselh(v,x) is a synonym for bessel(v,1,x) and besselh[k](v,x) is equivalent to besselh(v,k,x).
 • By default, these functions will evaluate only when the result is an exact value, or when the input x is a floating point number.  When x is a symbolic expression, they will remain in function form so that they can be manipulated symbolically by themselves or as part of a larger expression.

Examples

 > $\mathrm{with}\left(\mathrm{MTM}\right):$
 > $\mathrm{diff}\left(\mathrm{besselj}\left(v,x\right),x\right)$
 ${-}{\mathrm{BesselJ}}{}\left({v}{+}{1}{,}{x}\right){+}\frac{{v}{}{\mathrm{BesselJ}}{}\left({v}{,}{x}\right)}{{x}}$ (1)

Compatibility

 • The MTM[besseli], MTM[besselj], MTM[besselk], MTM[bessely] and MTM[besselh] commands were updated in Maple 2021.