On a certain method for finding zeros of analytic functions and its application to solving boundary value problems (Q1921854)

Let \(D\subset\mathbb{C}\) be a bounded \((n+1)\)-connected domain. We consider the equation \(\alpha(z)=0\), \(z\in D\), where \(\alpha\) is a function analytic in \(D\) and \(\alpha(z)\neq 0\) for all \(z\in \partial D\). The main result of the work is the following theorem: Let \[ m=(2\pi i)^{-1} \int_{\partial D}\alpha'(t)[\alpha(t)]^{-1}dt. \] Then the roots of the equation coincide with roots of the following equation: \[ c_0(z-z_0)^m+c_1(z-z_0)^{m-1}+\cdots+c_m= 0, \] where \(z_0\in D\) is arbitrary fixed, \(c_0=1\), \[ c_k=(-k)^{-1} \sum^{k-1}_{j=0} c_ja_{k-j},\quad a_k={1\over 2\pi i}\int_{\partial D} (t-z_0)^k\alpha'(t)[\alpha(t)]^{-1}dt. \]