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Expressions for Divergent Integrals in Terms of Convergent Ones

  • Henry E. Fettis
Chapter
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)

Abstract

This paper considers “divergent” integrals of the type
$${F_n}({z_0}) = \int\limits_a^b {\frac{{f(z)\;dz}}{{{{(z - {z_0})}^n}}}} $$
(1)
where n is an integer, and a < zO < b. For n = 1, the above integral is commonly defined by the “Cauchy principal value”:
$${F_1}({z_0}) = \mathop {\lim }\limits_{\varepsilon \to 0} \left[ {\int\limits_a^{{z_0} - \varepsilon } {\frac{{f(z)dz}}{{z - {z_0}}} + \int\limits_{{z_{0 + \varepsilon }}}^b {\frac{{f(z)dz}}{{z - {z_0}}}} } } \right]. $$
(2)

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References

  1. 1.
    Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, New Haven, London, 1923, and Dover, 1952.Google Scholar
  2. 2.
    Cheng, H.K., Meng, S.Y., and Smith, R.G., Transonic Swept Wings Studied by the Lifting Line Theory, AIAA Journal, 19 (1981), pp. 961–968.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • Henry E. Fettis

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