# Cauchy integral formula

There are a collection of results and theorems in complex analysis attributed to Cauchy.

These include, but are not limited to the Cauchy's theorem and the Cauchy integral theorems.

## Cauchy's theorem

Most commonly the reference to Cauchy's theorem is the following:

Given a complex valued function $f(z)=u(x,y)+iv(x,y)$ of the complex variable $z=x+iy$ analytic in some region ${\mathcal {R}}$ of the complex $z$ plane, and for which $f(z)$ is analytic both inside and on the simple closed curve $C$ inside ${\mathcal {R}}$ then

$\int _{C}f(z)dz=0.$ ## Cauchy integral theorem

The Cauchy Integral theorem states that for a function $f(z)$ which is analytic inside and on a simple closed curve $C$ in some region ${\mathcal {R}}$ of the complex $z$ plane, for a complex number $a$ inside $C$ $f(a)={\frac {1}{2\pi i}}\int _{C}{\frac {f(z)}{z-a}}dz.$ ## Cauchy integral formulas

The Cauchy Integral formulas for a complex valued function $f(z)$ which is analytic inside and on a simple closed curve $C$ in some region ${\mathcal {R}}$ of the complex $z$ plane, for a complex number $a$ inside $C$ $f^{(n)}(a)={\frac {n!}{2\pi i}}\int _{C}{\frac {f(z)}{(z-a)^{n+1}}}dz.$ 