Weierstrass double series theorem

If the complex functions$f_{0},\,f_{1},\,f_{2},\,\ldots$  are holomorphic in the disc  $|z-z_{0}|  and thus

 $\displaystyle f_{n}(z)=\sum_{\nu=0}^{\infty}a_{n\nu}(z-z_{0})^{\nu},\quad a_{n% \nu}=\frac{f_{n}^{(\nu)}(z_{0})}{\nu!}\quad\forall\,n,\,\nu$ (1)

in this disc, and if the function series

 $\displaystyle\sum_{n=0}^{\infty}f_{n}=f_{0}+f_{1}+f_{2}+\ldots$ (2)

converges uniformly to the function $F$ in each disc  $|z-z_{0}|\leqq\varrho$  where  $0<\varrho,  then also all the series

 $\displaystyle\sum_{n=0}^{\infty}a_{n\nu}=a_{0\nu}+a_{1\nu}+a_{2\nu}+\ldots% \quad(\nu=0,\,1,\,2,\,\ldots)$ (3)

converge, and in the disc  $|z-z_{0}|  one has

 $\displaystyle F(z)=\sum_{\nu=0}^{\infty}A_{\nu}(z-z_{0})^{\nu}$ (4)

where the $A_{\nu}$s are the sums of the series (3).

Proof.  Apparently, the series (2) converges uniformly also in every closed sub-disc of the open disc   $|z-z_{0}|.  Therefore the theorem 2 in the entry “theorems on complex function series (http://planetmath.org/TheoremsOnComplexFunctionSeries)” says that the sum $F(z)$ is holomorphic in  $|z-z_{0}|  and

 $F^{(\nu)}(z)=f_{0}^{(\nu)}(z_{0})+f_{1}^{(\nu)}(z_{0})+f_{2}^{(\nu)}(z_{0})+% \ldots\quad(\nu=0,\,1,\,2,\,\ldots).$

Theorem 3 in the same entry thus guarantees that $F(z)$ has the Taylor expansion  of the form (4) wherein

 $A_{\nu}=\frac{1}{\nu!}F^{(\nu)}(z_{0})\quad(\nu=0,\,1,\,2,\,\ldots).$

According to theorem 2 in the same entry the series (2) may be differentiated termwise,

 $A_{\nu}=\frac{1}{\nu!}\sum_{n=0}^{\infty}f_{n}^{(\nu)}(z_{0})=\sum_{n=0}^{% \infty}\frac{1}{\nu!}f_{n}^{(\nu)}(z_{0})=\sum_{n=0}^{\infty}a_{n\nu}$

Q.E.D.

 $\begin{array}[]{l}F(z)=f_{0}(z)+f_{1}(z)+\ldots+f_{n}(z)+\ldots=\\ =[a_{00}+a_{01}(z-z_{0})+a_{02}(z-z_{0})^{2}+\ldots+a_{0\nu}(z-z_{0})^{\nu}+% \ldots]\\ \,+[a_{10}+a_{11}(z-z_{0})+a_{12}(z-z_{0})^{2}+\ldots+a_{1\nu}(z-z_{0})^{\nu}+% \ldots]\\ \,+[a_{20}+a_{21}(z-z_{0})+a_{22}(z-z_{0})^{2}+\ldots+a_{2\nu}(z-z_{0})^{\nu}+% \ldots]\\ \qquad\qquad\ldots\ldots\\ \,+[a_{n0}+a_{n1}(z-z_{0})+a_{n2}(z-z_{0})^{2}+\ldots+a_{n\nu}(z-z_{0})^{\nu}+% \ldots]\\ \underline{\qquad\qquad\cdots\cdots\qquad\qquad\qquad\qquad\qquad\qquad\qquad% \qquad\qquad\qquad}\\ =A_{0}+A_{1}(z-z_{0})+A_{2}(z-z_{0})^{2}+\ldots+A_{\nu}(z-z_{0})^{\nu}+\ldots% \end{array}$
Title Weierstrass double series theorem WeierstrassDoubleSeriesTheorem 2013-03-22 16:48:15 2013-03-22 16:48:15 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 30B10 msc 40A05 msc 30D30 TheoremsOnComplexFunctionSeries