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A Generalization of the Cauchy Principal Value

Published online by Cambridge University Press:  20 November 2018

Charles Fox
Charles Fox*
Affiliation:
McGill University
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If a < u < b and n > 0 then

(1)

is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely

(2).

Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 110 - 117
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Hadamard, J., Lectures on Cauchy's Problem in Linear Partial Differential Equations (New York, 1952).Google Scholar
2. Muskhelishvili, N. I., Singular Integral Equations, Translated from Russian into English by J. R. M. Radok (Groningen, 1953).Google Scholar
3. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Translated from Russian into English by J. R. M. Radok (Groningen, 1953).Google Scholar
4. Titchmarsh, E. C., An Introduction to the Theory of Fourier Integrals (Oxford, 1937).Google Scholar