A multidimensional Jordan residue lemma with an application to Mellin-Barnes integrals

  • Mikael Passare
  • August Tsikh
  • Oleg Zhdanov
Part of the Aspects of Mathematics book series (ASMA, volume E 26)


The classical Jordan lemma states that if a function ψ is continuous on the real axis with a holomorphic continuation to the upper half plane Π+ = { z = x + iy;y > 0} except for a finite number of points {a} ⊂ Π+, and if ψ(z) tends to zero as |z| → ∞ in the closed half plane Π+, then
$$ \int_{ - \infty }^\infty {\psi (x){e^{iax}}dx = 2\pi i\sum\limits_{\left\{ a \right\}} {re{s_a}\;\omega ,} } $$
where α is an arbitrary positive number and ω denotes the differential form ψ(z)e iaz dz.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Griffiths and J. Harris, Principles of algebraic geometry, Pure Appl. Math., John Wiley and Sons, New York, 1978.Google Scholar
  2. [2]
    O. Marichev, Handbook of integral transforms of higher transcendental functions: Theory and algorithmic tables, Ellis Horwood, Chichester, 1983.zbMATHGoogle Scholar
  3. [3]
    M. Passare and A. Tsikh, O svjazjah meldu lokal'noj strukturoj golomorfnyh otobraenij, mnogomernymi vycetami i oboblcennymi preobrazovanijami Mellina, Dokl. Akad. Nauk, 325:4 (1992), 664-667. (English version will appear in Soviet Math. Dokl.)Google Scholar
  4. [4]
    M. Passare and A. Tsikh, Residue integrals and their Mellin transforms, Preprint TRITA/MAT92-0008, Royal Institute of Technology, Stockholm, 1992.Google Scholar
  5. [5]
    A. Tsikh, Metody teorii mnogomernyh vycetov, Doctoral dissertation, Novosibirsk, 1990.Google Scholar
  6. [6]
    A. Tsikh, Multidimensional residues and their applications, Transl. Math. Monographs 103, Amer. Math. Soc., Providence, 1992.Google Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1994

Authors and Affiliations

  • Mikael Passare
  • August Tsikh
  • Oleg Zhdanov

There are no affiliations available

Personalised recommendations