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Analytic Functions and Laurent Series

Any insight on where to go with these problems would be helpful and appreciative. This is my first time in complex analysis and I am having problems understanding the concepts. Seeing solutions to problems is helping me to understand how to approach other problems. Thank you very much!

I need help specifically on problems 2 and 10.
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Analytic Functions and Laurent Series are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.