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

A. P. Yuzhakov

Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 165-179

Résumé

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.