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1957•Canadian Journal of MathematicsOpen access

A Generalization of the Cauchy Principal Value

Charles Fox

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Abstract

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.

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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.

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Available abstract

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.

Key concepts: Mathematics, Cauchy principal value, Generalization, Hadamard transform, Infinity, Principal (computer security), Principal value, Value (mathematics)

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