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Two infinite integrals

Published online by Cambridge University Press:  20 January 2009

A. Erdélyi
A. Erdélyi
Affiliation:
The Mathematical Institute, 16 Chambers Street, Edinburgh, 1.
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1·1. The two functions F (λ, θ) and G (λ, θ), defined by the infinite integrals (1) and (2) respectively, below, occur in Kottler's theoretical discussion of the diffraction of a monochromatic plane wave by a perfectly black half plane. Some properties of these functions have been investigated by several recent writers.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 6 , Issue 2 , October 1939 , pp. 94 - 104
Copyright
Copyright © Edinburgh Mathematical Society 1939

References

page 94 note 1 Kottler, F., Ann. der Physik 71 (1923), 457–508 (496, 499).CrossRefGoogle Scholar

page 94 note 2 Copson, E. T. and Ferrar, W. L., Proc. Edin. Math. Soc. (2), 5 (1938), 159–68.CrossRefGoogle Scholar

page 94 note 3 Watson, G. N., Proc. Edin. Math. Soc. (2), 5 (1938), 173–81.CrossRefGoogle Scholar

page 94 note 4 Erdélyi, A., Proc. Edin. Math. Soc. (2), 6 (1939), 11.CrossRefGoogle Scholar

page 96 note 1 Watson, G. N., Theory of Bessel Functions (Cambridge, 1922), §222 (3), (4). This book will be referred to in the sequel as Bessel Functions.Google Scholar

page 97 note 1 Bessel Functions, §3·61 (5).

page 97 note 2 Bateman, H., Proc. London Math. Soc. (2), 3 (1905), 111–23 (120). See also Bessel Functions, §12·2 (3).CrossRefGoogle Scholar

page 98 note 1 Bessel Functions, §352.

page 99 note 1 Bessel Functions, § 1074 (5). We have used further

page 99 note 2 Bessel Functions, § 10·71 (3).

page 99 note 3 Bessel Functions, § 10·7 (10).

page 100 note 1 Bessel Functions, § 10·4 (2).

page 101 note 1 Bessel Functions, §10·4(5).

page 101 note 2 Pp. 666–97.

page 101 note 3 Third Edition (1938), 218–23.

page 101 note 4 Bessel Functions, § 10·72 (1).

page 102 note 1 Whittaker, E. T. and Watson, G. N., Modern Analysis (Cambridge, 1927), §16·12. To compare (25) and (26) put t = uz.Google Scholar

page 103 note 1 See also Modern Analysis, § 16·2.

page 103 note 2 Modem Analysis, § 16·2.