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A series of “cut” Bessel functions

Published online by Cambridge University Press:  20 January 2009

E. T. Copson  and
W. L. Ferrar
E. T. Copson
Affiliation:
University College, Dundee.
W. L. Ferrar
Affiliation:
Hertford College, Oxford.
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In Kottler's theoretical discussion1 of the diffraction of a plane wave of monochromatic light of wave-length 2π/k by a black halfplane, the function

where (r, θ, z) are cylindrical coordinates, plays an important part. In particular it is necessary to have asymptotic formulae for f (r, θ), valid when r is either very large or very small compared with the wave-length.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 3 , February 1938 , pp. 159 - 168
Copyright
Copyright © Edinburgh Mathematical Society 1938

References

page 159 note 1 Ann. der Phys. 71 (1923), 457–508 (in particular, pages 496 and 499). We have found it convenient to change the sign of i throughout.Google Scholar

page 159 note 2 Watson, G. N., A treatise on the theory of Bessel functions (Cambridge, 1922), 73. In later references, this work is cited as W.Google Scholar

page 163 note 1 The device used here was suggested by a more lengthy proof of the theorem on different lines.

page 166 note 1 Cf. the argument which gave equation (3.33).

page 167 note 1 See, for example, Titchmarsh, , Theory of Functions (Oxford, 1932), 423–4.Google Scholar

page 167 note 2 We have used here two well-known results in the theory of Fourier series. See, for example, Titchmarsh, , loc. dt., 440, Exx. 6, 7.Google Scholar