Calculus Small Changes, Big Ideas

Section 20.6 The residue theorem

• The coefficient of \((z-z_{0})\inv\) in the Laurent series of \(f\) is called the residue of \(f\) at \(z=z_{0}\text{,}\) and is denoted

  \begin{equation*} \resid{z=z_0}f(z)\text{.} \end{equation*}

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• Simple pole:

  \begin{equation*} \resid{z=z_0}f(z)=\limit{z\to z_0}(z-z_{0})f(z) \end{equation*}

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• Pole of order \(n\text{:}\)

  \begin{equation*} \resid{z=z_0}f(z)=\frac{1}{(n-1)!}\limit{z\to z_0}\dv[n-1]{z}\big((z-z_{0})f(z)\big) \end{equation*}

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• If \(f(z)=g(z)/h(z)\) and \(f\) has a simple pole at \(z=z_{0}\text{,}\) then

  \begin{equation*} \resid{z=z_0}f(z)=\dfrac{g(z_0)}{h'(z_0)} \end{equation*}

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• Theorem 20. Cauchy’s residue theorem.

  Let \(D\) be a simply connected domain and \(C\) a simple closed contour lying entirely within \(D\text{.}\) If \(f\) is meromorphic on and within \(C\) with isolated singularities \(\seq z1n\) within \(C\text{,}\) then

  \begin{equation*} \oint_{C} f(z)\dd{z}=2\pii\ii\sum_{k=1}^{n}\resid{z=z_0}f(z)\text{.} \end{equation*}

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• Some examples of real integrals evaluated using residues

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• Theorem 21. Argument principle.

  Let \(C\) be a simple closed contour lying entirely within a domain \(D\text{.}\) Suppose \(f\) is meromorphic in \(D\) and \(f(z)\neq 0\) on \(C\text{.}\) Then

  \begin{equation*} \frac{1}{2\pii\ii}\oint_{C}\frac{f'(z)}{f(z)}\dd{z}=N_{0}-N_{p} \end{equation*}

  where \(N_{0}\) is the total number of zeros of \(f\) (up to multiplicity) inside \(C\) and \(N_{p}\) is the number of poles of \(f\) (up to multiplicity) inside \(C\text{.}\)

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• Theorem 22. Rouché’s theorem.

  Let \(C\) be a simple closed contour lying entirely within a domain \(D\text{.}\) Suppose \(f\) and \(g\) are analytic in \(D\text{.}\) If the strict inequality \(|f(z)-g(z)|<|f(z)|\) holds for all \(z\) on \(C\text{,}\) then \(f\) and \(g\) have the same number of zeros (counted according to their order or multiplicities) inside \(C\text{.}\)

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  This can be used to prove the Fundamental theorem of algebra! THAT would be a great cherry on top!

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