# Cauchy Theorem¶

## 1. Simply and Multiply Connected Domains¶

Simply Connected Domain

A domain $$D\subseteq \mathbb{ C }$$ is simply connected if $$D$$ has no holes $$\Leftrightarrow C \setminus D$$ is connected.

Multiply Connected Domain

A domain that is not simply connected is called multiply connected.

## 2. Cauchy Theorem¶

Theorem: Let $$D \subseteq C$$ be simply connected domain and $$f:D\to \mathbb{C}$$ is any analytic function. Then for any closed contour $$\gamma$$ in $$D$$ , We have \begin{align}\int_\gamma f(z) dz=0\end{align}

Proof: Let $$f(z)= u(x,y) + iv(x,y)$$ and the contour

\begin{align} \gamma(t) &= x(t) + i y(t);& t\in [a,b]\\ \gamma'(t) &= x'(t) + i y'(t)& \end{align}

Now the integration will be

\begin{align} \int_\gamma f(z) dz &= \int_a^b f(\gamma(t))\gamma'(t)dt \\ &= \int_a^b (u(x,y) + iv(x,y)) (x' + iy') dt\\ &= \int_a^b (ux'-vy') dt + i\int_a^b(vx'+uy') dt \\ &= \oint_\gamma udx - vdy + i \oint_\gamma vdx +udy \end{align}

Now usign green theorem, we get the follwoing integral as

\begin{align} \phantom{\int_\gamma f(z) dz} &= \iint\limits_{R}(-v_x - u_y)dy dx +\iint\limits_{R}(u_x - v_y) dy dx \end{align}

Since $$f$$ is analytic, hence it will satisfy the CR-equation, $$i.e.,$$ $$u_x=v_y$$ and $$v_x=-u_y$$

\begin{align} \phantom{\int_\gamma f(z) dz} &= \iint\limits_{R}(-v_x - u_y)dy dx +\iint\limits_{R}(u_x - v_y) dy dx\\ &= 0. \end{align}

Ex: Find the following integration

1. $$\displaystyle \int_{\vert z \vert = 2}\frac{ e^z }{ z^2-9 }dz$$,
2. $$\displaystyle \int_{\vert z \vert = 2}\frac{ 2z+1 }{ e^z }dz$$.

Here we have to evaluate the integral on a circle of radius $$2$$, centerd at $$0$$.

For (1), the function $$\displaystyle f(z) = \frac{ e^z }{ z^2-9 }$$ is analytic in entire complex plane except $$z=\pm 3$$, but these points are outside the contour, hence by Cauchy theorem, the integration will be $$0$$.

For (2), the function $$\displaystyle f(z) = \frac{ 2z+1 }{ e^z }$$ is analytic in entire complex plane, hence by Cauchy theorem, the integration will be $$0$$.

Note: A function of the form of $$f/g$$ is analytic everywhere except $$g=0$$, provided $$f$$ and $$g$$ are analytic.