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Cauchy Principal Value

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The Cauchy principal value of an integral of a function f about a point c in the interior of the integration interval with a<c<b is given by

PVint_a^bf(x)dx=lim_(epsilon->0^+)[int_a^(c-epsilon)f(x)dx+int_(c+epsilon)^bf(x)dx]

when the limit exists (Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function f is defined by

PVint_(-infty)^inftyf(x)dx=lim_(R->infty)int_(-R)^Rf(x)dx

when the limit exists. The Cauchy principal value is also known as the principal value integral (Henrici 1988, p. 261). For a simple pole in the interior of the integration interval, it agrees with the corresponding Hadamard finite part. In general, however, the finite part and the Cauchy principal value are distinct notions (Galapon 2016).

The Cauchy principal value of an integral having no nonsimple poles can be computed in the Wolfram Language using Integrate[f, {x, a, b}, PrincipalValue -> True]. Cauchy principal values of functions with possibly nonsimple poles can be computed numerically using the "PrincipalValue" method in NIntegrate.

Cauchy principal values are important in the theory of generalized functions, where they allow extension of L^2 results to L^1.

Cauchy principal values are sometimes simply known as "principal values" (e.g., Vladimirov 1971, p. 75) even though they are not related to the principal value of complex analysis.

The most common designation for the Cauchy principal values seems to be PVintf(x)dx (Henrici 1988, pp. 259-262; Gradshteyn and Ryzhik 2000, p. 523). Sometimes, no explicit designation is used (Harris and Stocker 1998, p. 552; Gradshteyn and Ryzhik 2000, p. 248). Other notations include P (Arfken 1985, p. 403), P.V. (Apelblat 1983, p. viii), P (Morse and Feshbach 1953, p. 368; most Russian authors), Pv (Vladimirov 1971), (CPV) (Bronshtein and Semendyayev 1997, p. 282), and V.P. (Brychkov 1992, p. 7). For integrals with finite integration limits, the Cauchy principal value is sometimes denoted -int_a^bf(x)dx (Zwillinger 1995, p. 346).

See also

Finite Part, Hilbert Transform, Logarithmic Integral, Principal Value, Simple Pole

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References

Apelblat, A. Table of Definite and Indefinite Integrals. Amsterdam, Netherlands: Elsevier, 1983.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 401-403, 1985.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.Brychkov, Yu. A.; Glaeske, H.-J.; Prudnikov, A. P.; and Tuan, V. K. Multidimensional Integral Transformations. Philadelphia, PA: Gordon and Breach, 1992.Cauchy, A. "Sur un nouveau genre de calcul analogue au calcul infinitésimal." Exercises de mathematiques 1826. Reprinted in Oeuvres complètes, Ser. 2, Vol. 6. Paris, France: Gauthier-Villars, pp. 23-37, 1882-1974.Dieudonné, J. Geschichte der Mathematik 1700-1900: Ein Abriß. Berlin, Germany: VEB Deutscher Verlag der Wissenschaften, p. 149, 1985.Galapon, E. A. "The Cauchy Principal Value and the Hadamard Finite-Part Integral as Values of Absolutely Convergent Integrals." J. Math. Phys. 57, 033502, 2016. https://doi.org/10.1063/1.4943300.Gradshteyn, I. S. and Ryzhik, I. M. "The Principal Values of Improper Integrals." §3.05 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 248, 2000.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998.Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series, Integration, Conformal Mapping, Location of Zeros. New York: Wiley, 1988.Maurin, K. Analysis: Part Two: Integration, Distributions, Holomorphic Functions, Tensor and Harmonic Analysis. Amsterdam, Netherlands: Kluwer, 2001.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 1953.Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 158, 1991.Vladimirov, V. S. Equations of Mathematical Physics. New York: Dekker, 1971.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.

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Cauchy Principal Value

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Weisstein, Eric W. "Cauchy Principal Value." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchyPrincipalValue.html

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