theorems on complex function series

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

$f_1(z)+f_2(z)+f_3(z)+\mathrm{\dots }$(1)

converges uniformly on $\gamma$ to the sum function $F$, then one has

$${\int }_{\gamma }F(z)𝑑z={\int }_{\gamma }{f}_1(z)𝑑z+{\int }_{\gamma }{f}_2(z)𝑑z+{\int }_{\gamma }{f}_3(z)𝑑z+\mathrm{\dots }$$

Theorem 2.  If the functions  $f_1,f_2,f_3,\mathrm{\dots }$  are holomorphic in a domain $A$ and the series (1) converges uniformly in every closed (http://planetmath.org/ClosedSet) disc of $A$, then also the sum function $F$ of (1) is holomorphic in $A$ and the equality

${\displaystyle \frac{d^{n}F(z)}{d{z}^n}}=F^{(n)}(z)=f_1^{(n)}(z)+f_2^{(n)}(z)+f_3^{(n)}(z)+\mathrm{\dots }$(2)

is true for every positive integer $n$ in all points of $A$.  The series (2) converges uniformly in every compact subdomain of $A$.

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)

$$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{(z-z_0)}^n\mathit{\hspace{1em}}\mathrm{where}\mathit{\hspace{1em}}{a}_n=\frac{f^{(n)}(z_0)}{n!}\mathit{\hspace{1em}}(n=\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }).$$

This is valid at least in the greatest disk   which contains points of $A$ only.