Theorem 1. If the complex functions $f_1,\,f_2,\,f_3,\,\ldots$ are continuous on the path $\gamma$ and the series

(1)

Theorem 2. If the functions $f_1,\,f_2,\,f_3,\,\ldots$ are holomorphic in a domain $A$ and the series (1) converges uniformly in every closed disc of $A$ , then also the sum function $F$ of (1) is holomorphic in $A$ and

(2)

Theorem 3. If $f(z)$ is holomorphic in a domain $A$ and $z_0$ is a point of $A$ , then one can expand $f(z)$ to a power series (the so-called Taylor series) $$f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n \quad \mathrm{where} \quad a_n = \frac{f^{(n)}(z_0)}{n!}\quad(n = 0,\,1,\,2,\,\ldots).$$ This expansion is valid at least in the greatest disk $|z-z_0| < r\, (\leqq \infty)$ which contains points of $A$ only.