Integral Transforms and their Applications pp 237-253 | Cite as

# Hankel Transforms

Chapter

## Abstract

Bessel functions have frequently occurred in our investigations of the Laplace and Fourier transforms; indeed, we could rewrite most of the formulas we have derived in terms of Bessel functions of order ±1/2, since (2x/π)^{1/2}K_{1/2} (x) = exp(−x), with similar relations for sin(x) and cos(x).

## Keywords

Bessel Function Hankel Function Thin Elastic Plate Fourier Cosine Initial Temperature Distribution
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## Footnotes

- 1.For example, Sneddon (1972).Google Scholar
- 2.Lommnl’s integral is for any pair of cylinder functions U
_{ν}and V_{ν}[Watson (1958), p. 134]. It may be used to obtain results such asGoogle Scholar - 3.If F
_{ν}(k) is analytic in a region of the complex plane containing a ≤ k ≤ b, then we replace (3) byGoogle Scholar - 4.In particular the case b → ∞ is easy to handle. Also, if the interval 0 ≤ x ≤ ∞ can be split up into a finite number of subintervals in each of which the condition of MacRobert’s proof applies, then the proof is easily generalized. This covers most functions which arise in applications.Google Scholar
- 5.We have chosen the constants 2π in a more symmetrical way than in (11.1) and (11.2).Google Scholar
- 6.Another transform is obtained from the choice.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1985