Geodesic
An operator generalization of the logarithmic residue theorem and the theorem of Rouché
Sbornik. Mathematics, Tome 13 (1971) no. 4, pp. 603-625 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain the operator generalization of the theorem on the logarithmic residue for meromorphic operator-functions. The proof of the generalization is based on a theorem concerning a special factorization of a meromorphic operator-function at a point. This theorem also allows us to generalize, to the case of meromorphic operator-functions, the formula of M. V. Keldysh for the principal part of the resolvent as well as several other theorems. A definition is given for the multiplicity of a pole for a meromorphic operator-function. The basic properties of the multiplicity of a pole are proved, and also a generalization of the Rouché theorem. Bibliography: 16 titles. 
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     title = {An operator generalization of the logarithmic residue theorem and the theorem of {Rouch\'e}},
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I. Ts. Gokhberg; E. I. Sigal. An operator generalization of the logarithmic residue theorem and the theorem of Rouché. Sbornik. Mathematics, Tome 13 (1971) no. 4, pp. 603-625. http://geodesic.mathdoc.fr/item/SM_1971_13_4_a7/

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