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Weierstrass's Double Series Theorem

Let all of the functions

f_n(z)=sum_(k=0)^inftya_k^((n))(z-z_0)^k
(1)

with n=0, 1, 2, ..., be regular at least for |z-z_0|<r, and let

F(z)=sum_(n=0)^(infty)f_n(z)
(2)
=[a_0^((0))+a_1^((0))(z-z_0)+...+a_k^((0))(z-z_0)^k+...]+[a_0^((1))+a_1^((1))(z-z_0)+...+a_k^((1))(z-z_0)^k+...]+...+[a_0^((n))+a_1^((n))(z-z_0)+...+a_k^((n))(z-z_0)^k+...]+...
(3)

be uniformly convergent for |z-z_0|<=rho<r for every rho<r. Then the coefficients in any column form a convergent series. Furthermore, setting

a_k^((0))+a_k^((1))+...+a_k^((n))+...=sum_(n=0)^inftya_k^((n))=A_k
(4)

for k=0, 1, 2, ..., it then follows that

sum_(k=0)^inftyA_k(z-z_0)^k
(5)

is the power series for F(z), which converges at least for |z-z_0|<r.

See also

Double Series

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References

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 83, 1996.

Referenced on Wolfram|Alpha

Weierstrass's Double Series Theorem

Cite this as:

Weisstein, Eric W. "Weierstrass's Double Series Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrasssDoubleSeriesTheorem.html

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