theorems on complex function series

Theorem 1.  If the complex functions$f_{1},\,f_{2},\,f_{3},\,\ldots$  are continuous on the path $\gamma$ and the series

 $\displaystyle f_{1}(z)+f_{2}(z)+f_{3}(z)+\ldots$ (1)

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

 $\int_{\gamma}F(z)\,dz\;=\;\int_{\gamma}f_{1}(z)\,dz+\int_{\gamma}f_{2}(z)\,dz+% \int_{\gamma}f_{3}(z)\,dz+\ldots$

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 (http://planetmath.org/ClosedSet) disc of $A$, then also the sum function $F$ of (1) is holomorphic in $A$ and the equality

 $\displaystyle\frac{d^{n}F(z)}{dz^{n}}\;=\;F^{(n)}(z)\;=\;f_{1}^{(n)}(z)+f_{2}^% {(n)}(z)+f_{3}^{(n)}(z)+\ldots$ (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 )

 $f(z)\;=\;\sum_{n=0}^{\infty}a_{n}(z\!-\!z_{0})^{n}\quad\mathrm{where}\quad a_{% n}\;=\;\frac{f^{(n)}(z_{0})}{n!}\quad(n\,=\,0,\,1,\,2,\,\ldots).$

This is valid at least in the greatest disk  $|z-z_{0}|  which contains points of $A$ only.

Title theorems on complex function series TheoremsOnComplexFunctionSeries 2013-03-22 16:47:55 2013-03-22 16:47:55 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 30B99 msc 40A30 IdentityTheoremOfPowerSeries WeierstrassDoubleSeriesTheorem