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- Jan 26, 2012
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Thanks to those who participated in last week's POTW!! Here's this week's problem!
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Problem: Let $f(z)$ be a complex valued function that has a pole of order $m\geq 1$ at $z=z_0$. Prove that the residue of $f$ at this point is given by
\[\text{res}(f,z_0)=\lim_{z\rightarrow z_0}\frac{1}{(m-1)!}\frac{\,d^{m-1}}{\,dz^{m-1}} \left[ (z-z_0)^m f(z) \right]. \]
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Hint:
Remember to read the POTW submission guidelines to find out how to submit your answers!
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Problem: Let $f(z)$ be a complex valued function that has a pole of order $m\geq 1$ at $z=z_0$. Prove that the residue of $f$ at this point is given by
\[\text{res}(f,z_0)=\lim_{z\rightarrow z_0}\frac{1}{(m-1)!}\frac{\,d^{m-1}}{\,dz^{m-1}} \left[ (z-z_0)^m f(z) \right]. \]
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Hint:
Consider the Laurent series characterization of poles for $f(z)$, i.e. let
\[f(z)=\frac{a_{-m}}{(z-z_0)^m}+\cdots+\frac{a_{-1}}{z-z_0}+a_0+a_1(z-z_0)+a_2(z-z_0)^2+\ldots\]
Find a way to extract the residue $a_{-1}$ from this.
\[f(z)=\frac{a_{-m}}{(z-z_0)^m}+\cdots+\frac{a_{-1}}{z-z_0}+a_0+a_1(z-z_0)+a_2(z-z_0)^2+\ldots\]
Find a way to extract the residue $a_{-1}$ from this.
Remember to read the POTW submission guidelines to find out how to submit your answers!