Question

Expand f(z) = \frac{1}{(z-1)^2(z-3))} in a Laurent series valid for i) 0 < \lvert z-1 \rvert < 2 and ii) 0 < \lvert z-3 \rvert < 2

07 Feb 2024
Answer :
Word Count : 255

To expand \( f(z) = \frac{1}{(z-1)^2(z-3)} \) into a Laurent series valid for the annuli \( 0 < \lvert z-1 \rvert < 2 \) and \( 0 < \lvert z-3 \rvert < 2 \), we'll use partial fraction decomposition and then expand each term separately.

First, let's decompose \( f(z) \) ____ _____ ___ ____ _____ ________ ____ _________ ___ ____.
________ ________ _______ ____ _________ _____ __________ _______.
_________ _____ ________ ________ ____ ______ _______ ______ ____ __________ _________.
____ __________ ______ _________ ________ ____ ________ ______.
___ ________ _______ _________ ____ ___ ____ ___ __________ _____ ______ ___.
__________ ______ _______ _____ ____ ___ ______.
________ ______ _______ ____ __________ _______ _____ __________ ______ _____ ________ _________.
___ ____ ______ _______ ____ ____ ______.
____ _________ _______ _____ _______ __________ __________ _______ ________.
________ __________ _________ _____ _________ _________ ________ _______ ______.
___ _____ _________ _________ _______.
____ _________ _____ _____ _____ _____.
__________ ______ __________ ___ _____ _____ ___ __________.
____ _______ ___ _____ ____.
______ _____ ___ __________ __________ ______ ____.
_____ ________ _______ ____ __________ ______ __________ _____.
_________ ___ ______ __________ ________ ____ _______ ____ _____.
_________ __________ ___ ______ _________ _____ _______.
____ ____ _______ _________ ______ _______ __________.
_______ _______ ________ ____ _____ ____ ___ ____ _________ ____ _____.
_________ _________ ________ _______ ___ ____ _____ _________.
______ ___ ___ _________ ________ _________ ______ ______ ___ ____ _______.
________ __________ ____ _____ __________ __________.
_______ _____ ______ _____ _________ ___ ________ _________.
________ __________ _____ _______ ______.
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