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XVI.—Fourier Integrals

Published online by Cambridge University Press:  15 September 2014

T. M. MacRobert
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Extract

1. Fourier's Integral Theorem.—In a former paper it was shown that certain formulæ, usually obtained by means of Fourier's Integral Theorem, could also be proved by contour integration: and, indeed, Fourier's Integral Theorem can be established by this method. The functions involved must be holomorphic, so that the proof does not justify the application of the theorem to such wide classes of functions as the usual proof by means of Dirichlet Integrals. In what follows it will be assumed that all the integrals are convergent, and the changes of order of integration justified. Fourier's Integral Theorem will be proved in the following form.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 51 , 1932 , pp. 116 - 126
Copyright
Copyright © Royal Society of Edinburgh 1932

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References

page 116 note * Proc. Edin. Math. Soc., ser. 2, 2 (1929), 26.

page 118 note * Proc. Lond. Math. Soc., 35 (1902), 428.

page 118 note † G.M.M., p. 23. [The letters G.M.M. refer to Bessel Functions by Gray, Mathews, and MacRobert.] It may be noted that 2Gn(z) = iπHn(1)(z).

page 118 note ‡ G.M.M., p. 57.

page 118 note ║ G.M.M., pp. 57, xiv.

page 118 note § G.M.M., pp. 57, xiv.

page 118 note ¶ G.M.M., p. 69.

page 120 note * Proc. Roy. Soc. Edin., 42 (1922), 90.

page 122 note * G.M.M., p. 69.

page 122 note † G.M.M., p. 76, ex. 14, p. 256 (35).

page 123 note * Proc. Edin. Math. Soc., 14 (1923), 88.

page 123 note † Cf. the author's Functions of a Complex Variable, pp. 248, 249; also Proc. Edin. Math. Soc., 37 (1919), 81, 84.