Cauchy-Goursat Theorem

Theorem
Let $U$ be a simply connected open subset of the complex plane $\C$.

Let $\gamma : \left[{a \,.\,.\, b}\right] \to U$ be a closed contour in $U$.

Let $f: U \to \C$ be holomorphic in $U$.

Then:
 * $\displaystyle \oint_\gamma f \left({z}\right) \ \mathrm d z = 0$

Step 1
Let $C_1$ and $C_2$ be two contours such that:


 * $\gamma := C_1 + \left({- C_2}\right)$


 * $C_1$ has domain $\left[{a_1 \,.\,.\, b_1}\right]$

and:
 * $C_2$ has domain $\left[{a_2 \,.\,.\, b_2}\right]$

Then:


 * $C_1 \left({a_1}\right) = C_2 \left({a_2}\right)$

and


 * $C_1 \left({b_1}\right) = C_2 \left({b_2}\right)$

Thus:

Example
Let $\gamma \left({t}\right) = e^{i t}$.

Give $\gamma$ the domain $\left[{0, 2 \pi}\right)$.

Now, let $f \left({z}\right) = z^2$. Then,

Also known as
This result is also known as Cauchy's Integral Theorem or the Cauchy Integral Theorem.