10.6.17 Analytic functions

Problem: a) Point \( z_{0} \in \mathbb{C} \) is a zero of multiplicity \( m \) of the analytic function in \( z_{0} \) of function \( f(z) \) only if \( f(z) \) is represented as \( f(z)=\left(z-z_{0}\right)^{m} \cdot \varphi(z) \), where \( \varphi(z) \), is analytic in \( z_{0} \), and \( \varphi\left(z_{0}\right) \neq 0 \). b) Let the zeros of function \( f(z) \) be \( z_{1}, z_{2}, \ldots, z_{n} \in \mathbb{C} \), with orders \( m_{1}, m_{2}, \ldots, m_{n} \in \mathbb{N} \), respectively, and \( f(z) \) is analytic in points \( z_{1}, z_{2}, \ldots, z_{n}\left(z_{i} \neq z_{j}, i \neq j\right) \). Then \( f(z) \) is represented in the form of \( f(z)=\left(z-z_{1}\right)^{m_{1}}(z- \) \( \left.-z_{2}\right)^{m_{2}} \cdot \ldots \cdot\left(z-z_{n}\right)^{m_{n}} \cdot \varphi(z) \), where \( \varphi\left(z_{i}\right) \neq 0, i=\overline{1, n} \) and \( \varphi(z) \) is analytic in points \( z_{1}, \ldots, z_{n} \).