Geodesic
On an application of the multiple logarithmic residue to the expansion of implicit functions in power series
Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 165-179 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

By means of a multidimensional analog of the theorem of logarithmic residues, representations are found for the implicit functions $z_j=\varphi_j(w)$, $j=1,\dots,n$, defined by the system of equations $$ F_j(w,z)=0,\qquad j=1,\dots,n, $$ where $w=(w_1,\dots,w_m)$, $z=(z_1,\dots,z_n)$, $F_j(0,0)=0$, and $\frac{\partial(F_1,\dots,F_n)}{\partial(z_1,\dots,z_n)}\big|_{(0,0)}\ne0,$ as also for the function $\Phi(w,z)=\Phi(w,\varphi(w))$, $\varphi=(\varphi_1,\dots,\varphi_n)$, where $F_1,\dots,F_n$ 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. 
@article{SM_1975_26_2_a1,
     author = {A. P. Yuzhakov},
     title = {On~an application of~the multiple logarithmic residue to the expansion of implicit functions in power series},
     journal = {Sbornik. Mathematics},
     pages = {165--179},
     year = {1975},
     volume = {26},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1975_26_2_a1/}
}
TY  - JOUR
AU  - A. P. Yuzhakov
TI  - On an application of the multiple logarithmic residue to the expansion of implicit functions in power series
JO  - Sbornik. Mathematics
PY  - 1975
SP  - 165
EP  - 179
VL  - 26
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1975_26_2_a1/
LA  - en
ID  - SM_1975_26_2_a1
ER  - 
%0 Journal Article
%A A. P. Yuzhakov
%T On an application of the multiple logarithmic residue to the expansion of implicit functions in power series
%J Sbornik. Mathematics
%D 1975
%P 165-179
%V 26
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1975_26_2_a1/
%G en
%F SM_1975_26_2_a1
A. P. Yuzhakov. On an application of the multiple logarithmic residue to the expansion of implicit functions in power series. Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 165-179. http://geodesic.mathdoc.fr/item/SM_1975_26_2_a1/

[1] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, Fizmatgiz, Moskva, 1958

[2] E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza, Fizmatgiz, Moskva, 1963

[3] T. J. Stielties, “Sur une généralisation de la série de Lagrange”, Ann. Sci. Ecole Norm. Sup., 2 (1885), 93–98 | MR

[4] H. Poincaré, “Sur les rèsidus des intégrates doubles”, Acta Math., 9 (1887), 321–380 | DOI | MR

[5] E. Gursa, Kurs matematicheskogo analiza, t. II, ONTI, Moskva–Leningrad, 1936

[6] I J. Good, “Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes”, Proc. Cambridge Phil. Soc., 54:4 (1960), 367–380 | DOI | MR

[7] J. K. Percus, “A note on Extension of the Lagrange Inversion formula”, Comm. Pure and Appl. Math., 17:2 (1964), 137–146 | DOI | MR | Zbl

[8] R. A. Sack, “Interpretation of Lagrange's expansion and its generalization to several variables as integration formulas”, J. Soc. Indust. Appl. Math., 13:1 (1965), 47–59 | DOI | MR | Zbl

[9] R. A. Sack, “Generalization of Lagrange's expansion for functions of several implicity defined variables”, J. Soc. Indust. Appl. Math., 13:4 (1965), 913–926 | DOI | MR | Zbl

[10] A. Cayley, “Note sur une formule pour la reversion des séries”, J. reine and angew. Math., 52 (1856), 276–284 | Zbl

[11] A. Cayley, “Deuxième note sur une formule pour la reversion des séries”, J. reine and angew. Math., 54 (1857), 156–161 | Zbl

[12] J. J. Silvester, “On the change of systems of independent variables”, Quart. J. Math., 1 (1857), 43–56; 126–134

[13] E. E. Soltan, “O razlozhenii golomorfnykh funktsii v mnogomernye ryady Burmana-Lagranzha”, Sb. «Golomorfnye funktsii mnogikh kompleksnykh peremennykh», Krasnoyarsk, 1972, 129–137 | MR

[14] A. P. Yuzhakov, A. V. Kuprikov, “O logarifmicheskom vychete v $\mathbf{C}^n$”, Sb. «Nekotorye svoistva golomorfnykh funktsii mnogikh kompleksnykh peremennykh», Krasnoyarsk, 1973, 181–191

[15] Dzh. Milnor, Osobye tochki kompleksnykh giperpoverkhnostei, izd-vo «Mir», Moskva, 1971 | MR

[16] M. Erve, Funktsii mnogikh kompleksnykh peremennykh, izd-vo «Mir», Moskva, 1965 | MR