Complex Analysis: Use of Cauchy

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Equation "(31)" in the solution to problem 2 on page 39 is clarified by understanding the derivation of its second term, which initially caused confusion. The first term is straightforward, but the second term's origin was not immediately apparent to the poster. After some consideration, the poster resolved their uncertainty regarding the second term. This highlights the importance of careful analysis in complex analysis problems. The discussion emphasizes the learning process involved in tackling challenging mathematical concepts.
nateHI
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http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf

Please look at problem 2 on page 39 of the problems/solutions linked above.

I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of (31) is clear but I'm not sure where the 2nd term came from.
 
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nateHI said:
http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf

Please look at problem 2 on page 39 of the problems/solutions linked above.

I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of (31) is clear but I'm not sure where the 2nd term came from.

Nevermind I figured it out.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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