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 Mat. Sb. (N.S.), 1975, Volume 97(139), Number 2(6), Pages 177–192 (Mi msb3647)

This article is cited in 5 scientific papers (total in 5 papers)

On an application of the multiple logarithmic residue to the expansion of implicit functions in power series

A. P. Yuzhakov

Abstract: By means of a multidimensional analog of the theorem of logarithmic residues, representations are found for the implicit functions $z_j=\varphi_j(w)$, $j=1,…,n$, defined by the system of equations
$$F_j(w,z)=0,\qquad j=1,…,n,$$
where $w=(w_1,…,w_m)$, $z=(z_1,…,z_n)$, $F_j(0,0)=0$, and $\frac{\partial(F_1,…,F_n)}{\partial(z_1,…,z_n)}|_{(0,0)}\ne0,$ as also for the function $\Phi(w,z)=\Phi(w,\varphi(w))$, $\varphi=(\varphi_1,…,\varphi_n)$, where $F_1,…,F_n$ and $\Phi$ are holomorphic functions at $(0,0)\in\mathbf C_{(w,z)}^{m+n}$, in the form of power series and certain function series. In particular, a formula is obtained for the inverse of a holomorphic map in $\mathbf C^n$. One degenerate case is considered, where it is still possible to define single-valued branches of the implicit functions.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:2, 165–179

Bibliographic databases:

UDC: 517.55+517.522
MSC: Primary 32A05, 32A25; Secondary 32B99
Received: 08.07.1974

Citation: A. P. Yuzhakov, “On an application of the multiple logarithmic residue to the expansion of implicit functions in power series”, Mat. Sb. (N.S.), 97(139):2(6) (1975), 177–192; Math. USSR-Sb., 26:2 (1975), 165–179

Citation in format AMSBIB
\Bibitem{Yuz75} \by A.~P.~Yuzhakov \paper On~an application of~the multiple logarithmic residue to the expansion of implicit functions in power series \jour Mat. Sb. (N.S.) \yr 1975 \vol 97(139) \issue 2(6) \pages 177--192 \mathnet{http://mi.mathnet.ru/msb3647} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=399495} \zmath{https://zbmath.org/?q=an:0326.32002} \transl \jour Math. USSR-Sb. \yr 1975 \vol 26 \issue 2 \pages 165--179 \crossref{https://doi.org/10.1070/SM1975v026n02ABEH002475} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Hugo H. Torriani, “Constructive inverse function theorems”, Lett Math Phys, 13:4 (1987), 273
2. Hugo H. Torriani, “Constructive implicit function theorems”, Discrete Mathematics, 76:3 (1989), 247
3. Erich Berger, “Remarks on the Analytic Implicit Function Theorem”, Journal of Mathematical Analysis and Applications, 209:2 (1997), 435
4. R. Uitham, B.J. Hoenders, “The electromagnetic Brillouin precursor in one-dimensional photonic crystals”, Optics Communications, 281:23 (2008), 5910
5. V. R. Kulikov, V. A. Stepanenko, “On solutions and Waring's formulae for the system of $n$ algebraic equations with $n$ unknowns”, St. Petersburg Math. J., 26:5 (2015), 839–848
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