Cauchy-Goursat Theorem

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

Let $\partial D$ denote the closed contour bounding $D$.

Let $f: D \to \C$ be holomorphic everywhere in $D$.

Then:
 * $\ds \oint_{\partial D} \map f z \rd z = 0$

Proof
Begin by rewriting the function $f$ and differential $\rd z$ in terms of their real and imaginary parts:


 * $f = u + iv$


 * $\d z = \d x + i \rd y$

Then we have:


 * $\ds \oint_{\partial D} \map f z \rd z = \oint_{\partial D} \paren {u + iv} \paren {\d x + i \rd y}$

Expanding the result and again separating into real and imaginary parts yields two contour integrals of real variables:


 * $\ds \oint_{\partial D} \paren {u \rd x - v \rd y} + i \oint_{\partial D} \paren {v \rd x + u \rd y}$

We next apply Green's Theorem to each contour integral term to convert the contour integrals to surface integrals over $D$:


 * $\ds \iint_D \paren {-\dfrac {\partial v} {\partial x} - \dfrac {\partial u} {\partial y} } \rd x \rd y + \iint_D \paren {\dfrac {\partial u} {\partial x} - \dfrac {\partial v} {\partial y} } \rd x \rd y$

By the assumption that $f$ is holomorphic, it satisfies the Cauchy-Riemann Equations


 * $\dfrac {\partial v} {\partial x} + \dfrac {\partial u} {\partial y} = 0$
 * $\dfrac {\partial u} {\partial x} - \dfrac {\partial v} {\partial y} = 0$

The integrands are therefore zero and hence the surface integrals are zero.

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