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TurbulenceAn Introduction for Scientists and Engineers$

Peter Davidson

Print publication date: 2015

Print ISBN-13: 9780198722588

Published to Oxford Scholarship Online: August 2015

DOI: 10.1093/acprof:oso/9780198722588.001.0001

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Appendix 3 Hankel transforms and hypergeometric functions

Appendix 3 Hankel transforms and hypergeometric functions

Source:
Turbulence
Author(s):

P. A. Davidson

Publisher:
Oxford University Press

A3.1 Hankel transforms

(p.620) The table below refers to the Hankel transform pair

f(x)=0kf^(k)Jν(kx)dk,  f^(k)=0xf(x)Jν(kx)dx.

f(x)

fˆ(k)

xnepx2 ν+n+2>0, p>0

kvΓ(12v+12n+1)2v+1p1+(n+v)/2Γ(v+1)1F1(12v+12n+1,v+1,k2/4p)

xνepx2 ν>1, p>0

kν(2p)ν+1exp[ k2/4p ]

xνepx ν>1, p>0

2p2kνΓν+3/2p2+k2ν+3/2π1/2

x1epx2 ν>1, p>0

π1/22p1/2exp[ k2/8p ] Iν/2(k2/8p)

xnepx ν+n+2>0, p>0

(k/2p)νΓ(ν+n+2)pn+2Γ(ν+1)F(12ν+12n+1, 12ν+12n+32, ν+1, k2/p2)

(p.621)

f(x)

fˆ(k)

x1epx ν>1, p>0

kνp2+k21/2pνp2+k21/2

x1x2+a21/2 a>0, ν>1

Iν/2(ka/2) Kν/2(ka/2)

xn ν<n+2<3/2

2n+1Γ(12ν+12n+1)Γ(12ν12n)k(n+2)

xν(x2+a2)1+μ 1<ν<2μ+3/2

aνμkμ2μΓ(1+μ)Kνμ(ka)

Notation:

1F1=Kummer’s hypergeometric function, often written as M2F1=Gauss’ hypergeometric function, often written as FΓ=Gamma functionJν,Iν,Kν=Usual Bessel functions

A3.2 Hypergeometric functions

Kummer’s confluent hypergeometric function, M(a, b, x), sometimes written as 1F1(a;b;x), is a solution of Kummer’s equation,

xd2fdx2+(bx)dfdxaf=0.

It is

M(a,b,x)=1+axb+a2x2b22!++anxnbnn!+,

where

an=a(a+1)(a+2)(a+n1), a0=1.

Special cases of M are

M(a,a,x)=ex, M(1,2,2x)=x1exsinhx, M12,32,x2=π(2x)1erfx,

(p.622) and M satisfies the differential expressions

dndxnM(a,b,x)=anbnM(a+n,b+n,x),    xaddxM(a,b,x)=M(a+1,b,x)M(a,b,x).

For x we have

M(a,b,x)Γ(b)Γ(a)exxab,

and Kummer’s transformation rule tells us that

M(a,b,x)=exM(ba,b,x).

Gauss’ hypergeometric function, on the other hand, is

F(a,b,c,x)=2F1(a,b;c;x)=n=0anbncnxnn!.

Special cases of F are

F(1,1,2,x)=x1ln(1x),  F12,,12,32,x2=x1arcsinx,  F12,,1,32,x2=x1arctanx,

and F satisfies the differential expression

dndxnF(a,b,c,x)=anbncnF(a+n,b+n,c+n,x).