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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.

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Solution Summary

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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