Calculus Small Changes, Big Ideas

Section 20.4 Integrals of holomorphic functions

• LOTS of results in this chapter, but a lot of them could be proved as exercises, and the focus could be on how they make computations easier. The proofs essentially boil down to using the things we already used in the other sections, e.g. path independence if and only if closed paths give zero integral .

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• Theorem 15. Cauchy’s integral theorem.

  Suppose \(f\) is holomorphic in a simply connected domain \(D\) and that \(f'\) is continuous in \(D\text{.}\) Then for every simple closed contour \(C\) in \(D\text{,}\)

  \begin{equation*} \oint_{C} f(z)\dd{z}=0\text{.} \end{equation*}

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  Proof.

  Let \(\bv{F}=(u,-v)\) be the Pólya vector field of \(f\text{.}\)

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  \begin{align*} \oint_C f(z)\dd{z} \amp = \oint_C (u+\ii v)(\dd{x}+\ii\dd{y})\\ \amp = \oint_C (u\dd{x}-v\dd{y}) + \ii\oint_C (v\dd{x}+u\dd{y})\\ \amp = \oint_C \bv{F}\cdot\uv{t}\dd{s} + \ii\oint_C \bv{F}\cdot\uv{n}\dd{s}\\ \amp = \iint_D \Curl\bv{F}\dd{A} + \ii\iint_D \Div\bv{F}\dd{A}\\ \amp = \iint_D \qty(\pdv{u}{y}+\pdv{v}{x})\dd{A} + \ii\iint_D \qty(\pdv{u}{x}-\pdv{v}{y})\dd{A} \end{align*}

  Since \(f\) is holomorphic, it satisfies the Cauchy-Riemann equation, so \(\bv F\) is irrotational and incompressible. Hence

  \begin{equation*} \int_{C} f(z)\dd{z}=0\text{.} \end{equation*}

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• Theorem 16. Cauchy-Goursat theorem.

  Suppose \(f\) is holomorphic in a simply connected domain \(D\text{.}\) Then for every simple closed contour \(C\) in \(D\text{,}\)

  \begin{equation*} \oint_{C} f(z)\dd{z}=0\text{.} \end{equation*}

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  Much harder to prove! Does it based on triangular contours, then polygonal, then any.

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• Multiply connected domains lead to deformation of contours!

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  \begin{align*} \oint_{C\tsub{outside}} \bv{F}\cdot\dd{\bv{r}} - \oint_{C\tsub{inside}} \bv{F}\cdot\dd{\bv{r}} \amp = 0\\ \oint_{C\tsub{outside}} \bv{F}\cdot\dd{\bv{r}} \amp = \oint_{C\tsub{inside}} \bv{F}\cdot\dd{\bv{r}} \end{align*}
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• Theorem 17. Cauchy-Goursat for simply connected domains.

  Suppose \(C, C_{1}, \cdots, C_{n}\) are simple closed curves with positive orientation such that \(C_{1},C_{2},\cdots,C_{n}\) are interior to \(C\) but the regions interior to each \(C_{k}\text{,}\) \(k=1,2,\cdots,n\text{,}\) have no points in common. If \(f\) is holomorphic on each contour and at each point interior to \(C\) but exterior to all the \(C_{k}\text{,}\) \(k=1,2,\cdots,n\text{,}\) then

  \begin{equation*} \int_{C} f(z)\dd{z}=\sum_{k=1}^{n}\oint_{C_k}f(z)\dd{z}\text{.} \end{equation*}

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• Can also be shown to work for non-simply-connected contours.

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• Cauchy-Goursat implies that holomorphic functions also have path independence, and we can use antiderivatives!All the usual rules work too, like integration by parts. If \(F\) is an antiderivative of a continuous function \(f\text{,}\) then

  \begin{equation*} \int_{C} f(z)\dd{z}=F(b)-F(a) \end{equation*}

  Furthermore path independence of continuous \(f\) implies the existence of an antiderivative.

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• If \(f\) is holomorphic in a simply connected domain \(D\text{,}\) then it has an antiderivative.

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• Theorem 18. Cauchy’s integral formula.

  Suppose that \(f\) is holomorphic in a simply connected domain \(D\) and \(C\) is any simple closed contour lying entirely within \(D\text{.}\) Then for any point \(z_{0}\) within \(C\text{,}\)

  \begin{equation*} f(z_{0})=\frac{1}{2\pii\ii}\oint_{C}\frac{f(z)}{z-z_{0}}\dd{z}\text{.} \end{equation*}

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• Theorem 19. Cauchy’s integral formula for derivatives.

  Suppose that \(f\) is holomorphic in a simply connected domain \(D\) and \(C\) is any simple closed contour lying entirely within \(D\text{.}\) Then for any point \(z_{0}\) within \(C\text{,}\)

  \begin{equation*} f^{(n)}(z_{0})=\frac{n!}{2\pii\ii}\oint_{C}\frac{f(z)}{(z-z_{0})^{n+1}}\dd{z}\text{.} \end{equation*}

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• These formulas imply that the derivative of a holomorphic function is holomorphic, that is, all holomorphic functions are infinitely differentiable.

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