PlanetMath: Weierstrass double series theorem

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Weierstrass double series theorem

(Theorem)

If the complex functions $f_0,\,f_1,\,f_2,\,\ldots$ are holomorphic in the disc $\vert z-z_0\vert < r$ 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

in each disc $\vert z-z_0\vert \leqq \varrho$ where $0 < \varrho < r$ , 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 $\vert z-z_0\vert < r$ 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 $\vert z-z_0\vert < r$ . Therefore the theorem 2 in the entry “theorems on complex function series” says that the sum is holomorphic in $\vert z-z_0\vert < r$ and

$\displaystyle F^{(\nu)}(z) = f_0^{(\nu)}(z_0)+f_1^{(\nu)}(z_0)+f_2^{(\nu)}(z_0)+\ldots \quad (\nu = 0,\,1,\,2,\,\ldots).$

Theorem 3 in the same entry thus guarantees that

has the Taylor expansion of the form (4) wherein

$\displaystyle A_\nu = \frac{1}{\nu!}F^{(\nu)}(z_0) \quad (\nu = 0,\,1,\,2,\,\ldots).$

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

$\displaystyle A_\nu = \frac{1}{\nu!}\sum_{n=0}^\infty f_n^{(\nu)}(z_0) = \sum_{n=0}^\infty \frac{1}{\nu!}f_n^{(\nu)}(z_0) = \sum_{n=0}^\infty a_{n\nu}$

Q.E.D.

Note. In Weierstrass double series theorem it's a question of changing the summing order:

$\begin{displaymath} \begin{array}{l} F(z) = f_0(z)+f_1(z)+\ldots+f_n(z)+\ldots =... ...(z-z_0)+A_2(z-z_0)^2+\ldots+A_\nu(z-z_0)^\nu+\ldots \end{array}\end{displaymath}$

"Weierstrass double series theorem" is owned by pahio.

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See Also: theorems on complex function series

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Cross-references: Taylor expansion, open, closed, sums, converge, series, function, converges uniformly, function series, disc, holomorphic, complex functions
There are 2 references to this entry.

This is version 4 of Weierstrass double series theorem, born on 2007-03-06, modified 2007-03-19.
Object id is 9038, canonical name is WeierstrassDoubleSeriesTheorem.
Accessed 740 times total.

Classification:

AMS MSC:	40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
	30B10 (Functions of a complex variable :: Series expansions :: Power series )
	30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

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