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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
$z_j=\varphi_j(w)$,
$j=1,\dots,n$, 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.
UDC:
517.55+517.522
MSC: Primary
32A05,
32A25; Secondary
32B99 Received: 08.07.1974