MHB Problems for Cauchy Integral Formula

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The discussion focuses on challenges related to applying the Cauchy Integral Formula in specific problems. Participants suggest using partial fraction decomposition as a method to tackle these questions. Additionally, the use of Cauchy's differentiation formula is recommended for certain calculations. The conversation emphasizes the need for clarity on the appropriate techniques to solve the posed problems. Overall, the thread seeks guidance on effectively utilizing these mathematical tools.
shirokuma
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Hello everyone!
I am currently stuck at the two type of questions below, because I am not really sure what method should be used to calculate these question...
Could you give me a hint how to do these questions? :(
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Welcome, Shirokuma!

Consider using partial fraction decomposition. You might need to use Cauchy's differentiation formula at some points.
 

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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