Complex Analysis/Goursat's Lemma

Introduction

Goursat's Lemma is a crucial result in the proof of the Cauchy's integral theorem.It restricts the integration paths to triangles, making it provable via a geometric subdivision argument.

Statement

Let ${\displaystyle D\subseteq \mathbb {C} }$ be a closed triangle, ${\displaystyle G\supseteq D}$ an open set, and ${\displaystyle f\colon U\to \mathbb {C} }$ a holomorphic function. Then: ${\displaystyle \int _{\partial D}f(z),dz=0.}$

Proof

Set ${\displaystyle \Delta _{0}:=D}$. We inductively construct a sequence ${\displaystyle (\Delta _{n})_{n\geq 0}}$ with the properties:

1. ${\displaystyle \Delta _{n}\subseteq \Delta _{n-1}}$

2. ${\displaystyle {\mathcal {L}}(\partial \Delta _{n})=2^{-n}{\mathcal {L}}(\partial D)}$, where ${\displaystyle {\mathcal {L}}}$ represents the length of a curve

3. ${\displaystyle \left|\int _{\partial D}f(z),dz\right|\leq 4^{n}\left|\int _{\partial \Delta _{n}}f(z),dz\right|}$

For ${\displaystyle n\geq 0}$ and ${\displaystyle \Delta _{n}}$ already constructed, we subdivide ${\displaystyle \Delta _{n}}$ by connecting the midpoints of its sides, forming four subtriangles ${\displaystyle \Delta _{n+1}^{i}}$, ${\displaystyle 1\leq i\leq 4}$. Since the contributions of the midpoints cancel out in the integration, we have:

${\displaystyle {\begin{array}{rl}\displaystyle \left|\int _{\partial \Delta _{n}}f(z)\,dz\right|&=\displaystyle \left|\sum _{i=1}^{4}\int _{\partial \Delta _{n+1}^{i}}f(z)\,dz\right|\\&\leq \displaystyle \sum _{i=1}^{4}\left|\int _{\partial \Delta _{n+1}^{i}}f(z)\,dz\right|\\&\leq \displaystyle \max _{i}\left|\int _{\partial \Delta _{n+1}^{i}}f(z)\,dz\right|\end{array}}}$

Choose ${\displaystyle 1\leq i\leq 4}$ such that ${\displaystyle \left|\int _{\partial \Delta _{n+1}^{i}}f(z),dz\right|=\max _{i}\left|\int _{\partial \Delta _{n+1}^{i}}f(z),dz\right|}$ and set ${\displaystyle \Delta _{n+1}:=\Delta _{n+1}^{i}}$. Then, by construction: ${\displaystyle \Delta _{n+1}\subseteq \Delta _{n}}$, ${\displaystyle {\mathcal {L}}(\partial \Delta _{n+1})={\frac {1}{2}}{\mathcal {L}}(\partial \Delta _{n})=2^{-(n+1)}{\mathcal {L}}(\partial D)}$, and ${\displaystyle \left|\int _{\partial D}f(z),dz\right|\leq 4^{n}\left|\int _{\partial \Delta _{n}}f(z),dz\right|\leq 4^{n+1}\left|\int _{\partial \Delta _{n+1}}f(z),dz\right|.}$

This ensures ${\displaystyle \Delta _{n+1}}$ has the required properties.

Since all ${\displaystyle \Delta _{n}}$ are compact, ${\displaystyle \bigcap _{n\geq 0}\Delta _{n}\neq \emptyset }$. Let ${\displaystyle z_{0}\in \bigcap _{n\geq 0}\Delta _{n}}$. As ${\displaystyle f}$ is holomorphic at ${\displaystyle z_{0}}$, there exists a neighborhood ${\displaystyle V}$ of ${\displaystyle z_{0}}$ and a continuous function ${\displaystyle A\colon V\to \mathbb {C} }$ with ${\displaystyle A(z_{0})=0}$ such that: ${\displaystyle f(z)=f(z_{0})+(z-z_{0})f'(z_{0})+A(z)(z-z_{0}),\qquad z\in V.}$

Since the function ${\displaystyle z\mapsto f(z_{0})+(z-z_{0})f'(z_{0})}$ has a primitive, it follows for ${\displaystyle n\geq 0}$ with ${\displaystyle \Delta _{n}\subseteq V}$ that: ${\displaystyle \int _{\partial \Delta _{n}}f(z),dz=\int _{\partial \Delta _{n}}f(z_{0})+(z-z_{0})f'(z_{0})+A(z)(z-z_{0}),dz=\int _{\partial \Delta _{n}}A(z)(z-z_{0}),dz.}$

Thus, due to the continuity of ${\displaystyle A}$ and ${\displaystyle A(z_{0})=0}$, we obtain:

${\displaystyle {\begin{array}{rl}\displaystyle \left|\int _{\partial D}f(z)\,dz\right|&\leq \displaystyle 4^{n}\left|\int _{\partial \Delta _{n}}f(z)\,dz\right|\\&=\displaystyle 4^{n}\left|\int _{\partial \Delta _{n}}A(z)(z-z_{0})\,dz\right|\\&\leq \displaystyle 4^{n}\cdot {\mathcal {L}}(\partial \Delta _{n})\max _{z\in \partial \Delta _{n}}|z-z_{0}||A(z)|\\&\leq \displaystyle 4^{n}\cdot {\mathcal {L}}(\partial \Delta _{n})^{2}\max _{z\in \partial \Delta _{n}}|A(z)|\\&=\displaystyle {\mathcal {L}}(\partial D)\max _{z\in \partial \Delta _{n}}|A(z)|\\&\to \displaystyle {\mathcal {L}}(\partial D)|A(z_{0})|=0,\qquad n\to \infty .\end{array}}}$

Notation in the Proof

${\displaystyle \Delta _{n}}$ is the ${\displaystyle n}$-th subtriangle of the original triangle, with side lengths scaled by a factor of ${\displaystyle {\frac {1}{2^{n}}}}$.

${\displaystyle \partial \Delta _{n}}$ is the integration path along the boundary of the ${\displaystyle n}$-th subtriangle, with perimeter ${\displaystyle {\mathcal {L}}(\partial \Delta _{n})={\frac {1}{2^{n}}}\cdot {\mathcal {L}}(\partial \Delta _{0})}$.

See Also

• Goursat's Lemma (Details)
• rectifiable curve

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