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cbarker1

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MHB

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- TL;DR Summary
- How to compute partial fractions decompose when one the factors is a root of multiplicity in the residue method?

Dear Everybody, I am wondering how to compute the partial fraction decomposition of the following rational function: ##f(z)=\frac{z+2}{(z+1)^2(z^2+1)}.##

I understand how to do the simple poles of the function and how it is related to the decomposition's constants, i.e. ##f(z)=\frac{A_1}{z+i}+\frac{A_2}{z-i}+\frac{B_1}{z+1}+\frac{B_2}{(z+1)^2}##. Thus, I know that ##A_1=-\frac{i+2}{4}## and ##A_2=-\frac{-i+2}{4}.## But how do I computes double pole in terms of the residue, if possible? How I can write out ##B_1, B_2.##

I understand how to do the simple poles of the function and how it is related to the decomposition's constants, i.e. ##f(z)=\frac{A_1}{z+i}+\frac{A_2}{z-i}+\frac{B_1}{z+1}+\frac{B_2}{(z+1)^2}##. Thus, I know that ##A_1=-\frac{i+2}{4}## and ##A_2=-\frac{-i+2}{4}.## But how do I computes double pole in terms of the residue, if possible? How I can write out ##B_1, B_2.##