Word Count : 376

---

To find the Laurent series expansion of \( f'(z) = \frac{1}{(z-1)^2 (z-3)} \), we need to expand the function in two different regions: one around \( z = 1 \) and the other around \( z = 3 \). ### i) For \( 0 < |z - 1| < 2 \) In this region, we can treat \( z - 1 \) as small and use the fact that \( \frac{1}{z - 3} \) can be expanded around \( z = 1 \). First, express \( \frac{1}{z - 3} \) as a geometric series: _____ ____ _______ _____ _________ ____ __________.
_______ ________ _________ ________ _____ ___.
__________ _________ ___ _______ _______ ____ ________ ____ _____ ________ __________.
__________ ____ ____ _______ __________ _________ ________.
_______ ___ ____ ___ _________ ____ ______ ____ __________.
______ _______ _________ _________ ____ ___ ______ ________ ____ ________.
_____ ______ ____ __________ _____.
__________ _____ ____ ____ _________ _________ ________ __________ _____ ___ ____ _________.
__________ _________ _______ ______ __________ ______ ____ _____ _______ ______ __________ _______.
____ ___ __________ ______ ______ ____.
______ ______ ____ ______ __________ ____ ______ __________.
__________ ___ _______ ________ ________ __________ __________ ________.
_________ ___ ________ _____ __________ ____ _____ __________ __________ ___ ____.
__________ __________ ____ __________ ______ _____ ____ __________ ________ _______ _____.
__________ __________ ____ _______ __________ _____ __________ __________.
________ ____ _______ ______ ______ ____ _______ _______ __________ ___ __________ ____.
________ _______ _________ _______ ______ _________ __________ ____ _______ _____.
_________ ____ ____ _____ _________ _____ __________ ________ _______ _____ ___.
_____ _____ __________ ______ ______ ____.
____ __________ __________ ________ _________.
______ _________ __________ ______ _________ __________ _______ _______ _______ _______ ______.
________ ________ _______ ___ ____ ____ __________ ______ ________.
________ ________ _______ ________ _____ _______ __________ _____ __________ _______ _____.
________ ______ __________ ___ ______ ___ _________.
___ _________ __________ ______ _______ __________ _____ _____.
____ _________ _______ ___ _______ ______ ______ ______ _________ ______ ___ ____.
___ _______ _________ _____ _____.
______ _________ _______ ________ ________ ___ ________.
_______ ______ ________ _______ _____ _____.
_____ ______ ____ _________ ___ ______ __________ ____ _________ ________ ________.
____ ______ _______ _______ ____ __________ ____.
__________ ____ _________ ______ __________ ________ _______ _______ __________ ______ ______ ____.
_______.
Get Full Answer on WhatsApp