Mathematics Calculus and Analysis Real Analysis Integrals 46544
Integration of Uniformly Convergent Series : Cauchy's Theorem and Morera's Theorem
Prove the following:
Let be a sequence of functions continuous on a set
containing the contour , and suppose that converges uniformly to on .
Then the series converges to .
Using this result and the Generalized Cauchy
Integral formula for derivatives (see below),
show the following:
If all are analytic throughout a domain D and converges uniformly to
on any closed subdisk of , then the derivative exists and converges to
for every .
The Generalized Cauchy Integral formula for derivatives (no need to prove):
If is analytic inside and on the simple closed positively oriented contour
and if is any point inside , then
, n = 1,2,3,...
Please see the attached file for the fully formatted problems.
Integration of Uniformly Convergent Series, Cauchy's Theorem and Morera's Theorem are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.
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