Cauchy's Theorem Local Version

Subjects: Complex Analysis
Links: Contour Integrals in C, Analytic Functions

Goursat’s Theorem (Cauchy’s Theorem for Rectangles)

Let $f:\mathrm{\Omega }\subseteq \mathbb{C}\to \mathbb{C}$ be holomorphic on $\mathrm{\Omega }$ and $R=[a,b]\times [c,d]\subseteq \mathrm{\Omega }$. If $\mathrm{\Gamma }=\partial R$, and with $\gamma$ being a counterclockwise parametrization of $\mathrm{\Gamma }$, then

Local Version of Cauchy’s Theorem (Marsden)

Suppose $f:D\to \mathbb{C}$ is holomorphic on $D=B_{\rho }(z_0)\subseteq \mathbb{C}$, then

• $f$ has an primitive on $D$; that is, there is a function $F:D\to \mathbb{C}$ is holomorphic on $D$ and satisfies $F^{\prime }=f$ on $D$

• If $\mathrm{\Gamma }$ is a closed piecewise smooth curve in $D$ then

  $${\oint }_{\mathrm{\Gamma }}f=0$$

This tells us that if a function is holomorphic , then there’s a local primitive on an open neighborhood of $z_0$.

Lemma: Suppose $R$ is a rectangle with sides parallel to the axes, that $f$ is a function defined on an open set $G$ containing $R$, $f$ is holomorphic in $G\setminus \{z_1,\dots ,z_n\}$, and let $z_1,\dots ,z_m\notin \partial R$. Suppose that at $z_1$, the function $f$ satistisfies $\underset{z\to z_j}{lim}(z-z_j)f(z)=0$ for $1\le j\le n$. Then

This lemma holds under any of the following situations:

• $f$ is bounded in a deleted neighborhood of $z_1,\dots ,z_n$
• $f$ is continuous on $G$
• If $\underset{z\to z_1}{lim}f(z)$ exists

Lemma: Suppose $R$ is a rectangle with sides parallel to the axes, that $f$ is a continuous function defined on an open set $G$ containing $R$, $f$ is holomorphic in $G\setminus \{z_1,\dots ,z_n\}$. Then

Strengthened Local Version of Cauchy’s Theorem

Suppose $f:D\to \mathbb{C}$ , where $D=B_{\rho }(z_0)\subseteq \mathbb{C}$ and holomorphic on $D\setminus \{z_1,\dots ,z_n\}$ for some fixed points in $D$, with

• If $\gamma$ is a closed piecewise smooth curve in $D\setminus \{z_1,\dots ,z_n\}$ then

  $${\oint }_{\gamma }f=0$$