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Mat. Zametki, 1978, Volume 23, Issue 1, Pages 47–54 (Mi mz8117)  

A generalization of the inversion formulas of systems of power series in systems of implicit functions

V. A. Bolotova, A. P. Yuzhakovb

a Institute of Physics, Siberian Branch of USSR Academy of Sciences
b Krasnoyarsk State University

Abstract: Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functions $z_j=\varphi_j(w)=\varphi_j(w_1,…,w_m)$, $j=1,…,n$, defined by the system of equations $F_j(w,z)=F_j(w_1,…,2_m;z_1,…,z_n)=0$, $j=1,…,n$, $F_j(0,0)=0$, $\partial F_j(0,0)/\partial z_k=\delta_{jk}$ in a neighborhood of the point $(0,0)\in C_{(w,z)}^{m+n}$, in terms of the coefficients of the power series of the functions $F_j(w,z)$, $j=1,…,n$. As a corollary, well-known formulas are obtained for the inversion of multiple power series.

Full text: PDF file (568 kB)

English version:
Mathematical Notes, 1978, 23:1, 27–31

Bibliographic databases:

UDC: 517.5
Received: 26.04.1976

Citation: V. A. Bolotov, A. P. Yuzhakov, “A generalization of the inversion formulas of systems of power series in systems of implicit functions”, Mat. Zametki, 23:1 (1978), 47–54; Math. Notes, 23:1 (1978), 27–31

Citation in format AMSBIB
\Bibitem{BolYuz78}
\by V.~A.~Bolotov, A.~P.~Yuzhakov
\paper A~generalization of the inversion formulas of systems of power series in systems of implicit functions
\jour Mat. Zametki
\yr 1978
\vol 23
\issue 1
\pages 47--54
\mathnet{http://mi.mathnet.ru/mz8117}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=477105}
\zmath{https://zbmath.org/?q=an:0404.32001|0369.32001}
\transl
\jour Math. Notes
\yr 1978
\vol 23
\issue 1
\pages 27--31
\crossref{https://doi.org/10.1007/BF01104881}


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