Cauchy's Integral Theorem

Cauchy's Integral Theorem

Recall from the Cauchy's Integral Theorem for Rectangles page that if:

• $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$.
• $R \subseteq A$ is a rectangle contained in $A$ and whose interior is in $A$.
• $f$ is analytic on $A$

Then we have that:

(1)
\begin{align} \quad \int_R f(z) \: dz = 0 \end{align}

We will now state two other very important theorems. The first is called the Cauchy-Goursat Integral Theorem (for open disks), and the most general Cauchy Integral theorem. We will prove neither of these results as they're a bit technical but we will apply them extensively.

 Theorem 1 (The Cauchy-Goursat Integral Theorem for Open Disks): Let $f$ be be an analytic function on the open disk $D(z_0, r)$. Then for any closed, piecewise smooth curve $\gamma$ in $D(z_0, r)$ we have that $\displaystyle{\int_{\gamma} f(z) \: dz =0}$ Furthermore, there exists an analytic function $F$ on $D(z_0, r)$ for which $F'(z) = f(z)$.

For example, consider the the function $\displaystyle{f(z) = \frac{e^z}{z - 6}}$ and let $\gamma$ be the circle centered at the origin with radius $r$, i.e., $\gamma$ can be parameterized for $t \in [0, 2\pi)$ by:

(2)
\begin{align} \quad \gamma(t) = 2e^{it} \end{align}

Since $f$ is analytic everywhere except at $z = 6$ we see that $f$ is analytic on, say, $D(0, 3)$ and that $\gamma$ is a closed, piecewise smooth curve in $D(0, r)$.

So by the Cauchy-Goursat integral theorem for open disks we have that:

(3)
\begin{align} \quad \int_{\gamma} \frac{e^z}{z - 6} \: dz = 0 \end{align}

Before we state the full Cauchy's integral theorem we will need to go through some definitions. These definitions are not precise but will give us an idea of the context of Cauchy's integral theorem.

 Definition: Let $\gamma_1$ and $\gamma_2$ be two curves in some open set $A$. Then $\gamma_1$ is said to be Homotopic to $\gamma_2$ if $\gamma_1$ can be continuously deformed into $\gamma_2$. A piecewise smooth curve $\gamma$ in $A$ is said to be Homotopic to a Point $z_0 \in A$ if $\gamma$ can be continuously deformed to $z_0$.

Once again we must emphasize that this definition is not precise and a more rigorous approach would go into defining these terms more precisely. Nevertheless, hopefully the intuition behind these definitions is clear for curves in $\mathbb{C}$. We are now ready to state Cauchy's integral theorem.

 Theorem 2 (Cauchy's Integral Theorem): Let $A \subseteq \mathbb{C}$ be open and $f : A \to \mathbb{C}$. If $\gamma$ is a closed, piecewise smooth curve in $A$ that is homotopic to a point in $A$ then $\displaystyle{\int_{\gamma} f(z) \: dz =0}$.

Cauchy's integral theorem is remarkable as it simplifies many difficult integral problems.

For example let $\gamma$ be the unit circle ($\gamma(t) = e^{it}$ for $t \in [0, 2\pi )$) and consider the function $\displaystyle{f(z) = \sin (\cos (e^{z + z^2 + 5z^4}))}$. Notice that this function is analytic on all of $\mathbb{C}$ as it is a composition of functions analytic everywhere, and that $\gamma$ is a closed, piecewise smooth curve in $\mathbb{C}$ that is homotopic to a point in $\mathbb{C}$ (take any point $D(0, 1)$). So by Cauchy's integral theorem we have that:

(4)
\begin{align} \quad \int_{\gamma} f(z) \: dz = \int_{\gamma} \sin (\cos (e^{z + z^2 + 5z^4})) \: dz = 0 \end{align}