Purchase Solution

Integration of Uniformly Convergent Series : Cauchy's Theorem and Morera's Theorem

Not what you're looking for?

Ask Custom Question

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.

Purchase this Solution

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.

Purchase this Solution


Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Probability Quiz

Some questions on probability

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.