The Bessel functions of the first kind - Maple Help

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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 :

x2y''+xy'+v2+x2y=0

• 

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

x2y''+xy'v2+x2y=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

HankelH1v,x=BesselJv,x+IBesselYv,x

HankelH2v,x=BesselJv,xIBesselYv,x

Examples

BesselJ0,2

BesselJ0,2

(1)

evalf

0.2238907791

(2)

BesselK1,3.

0.0401564311312.41987883I

(3)

BesselI0,0

1

(4)

BesselY1.5+I,3.5I

0.95665185121.465483431I

(5)

seriesBesselJ3,x,x

148x31768x5+Ox6

(6)

xBesselJv,x

BesselJv+1,x+vBesselJv,xx

(7)

HankelH12.5,3.7+I

0.18092605720.08706107529I

(8)

xHankelH2v,x2

2HankelH2v+1,x2+vHankelH2v,x2x2x

(9)

convertHankelH2v,x,Bessel

BesselJv,xIBesselYv,x

(10)

convertAiryAix,Bessel

13xBesselI13,23x3x31/6+13x31/6BesselI13,23x3

(11)

convertKelvinKerv,x,BesselK

12BesselKv,12+12Ix2+ⅇ12Ivπ2BesselKv,1212Ix2ⅇ12Ivπ

(12)

See Also

Airy, Anger, BesselZeros, convert/Bessel, inifcns, inttrans[hankel], Kelvin, Struve


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