## 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

Source:
Turbulence
Publisher:
Oxford University Press

# A3.1 Hankel transforms

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

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$f(x)$

$fˆ(k)$

$xne−px2$

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

$xνe−px2$

$kν(2p)ν+1exp[ −k2/4p ]$

$xνe−px$

$2p2kνΓν+3/2p2+k2−ν+3/2π−1/2$

$x−1e−px2$

$xne−px$

(p.621)

$f(x)$

$fˆ(k)$

$x−1e−px$

$k−νp2+k21/2−pνp2+k21/2$

$x−1x2+a2−1/2$

$xn$ $−ν

$2n+1Γ(12ν+12n+1)Γ(12ν−12n)k−(n+2)$

$xν(x2+a2)1+μ$ $−1<ν<2μ+3/2$

$aν−μkμ2μΓ(1+μ)Kν−μ(ka)$

Notation:

# 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,

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It is

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where

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Special cases of M are

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(p.622) and M satisfies the differential expressions

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For $x→∞$ we have

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and Kummer’s transformation rule tells us that

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Gauss’ hypergeometric function, on the other hand, is

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Special cases of F are

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and F satisfies the differential expression

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