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

${\displaystyle \int _{C}f(z)dz=0.}$

## Cauchy integral theorem

The Cauchy Integral theorem states that for a function ${\displaystyle f(z)}$ which is analytic inside and on a simple closed curve ${\displaystyle C}$ in some region ${\displaystyle {\mathcal {R}}}$ of the complex ${\displaystyle z}$ plane, for a complex number ${\displaystyle a}$ inside ${\displaystyle C}$

${\displaystyle f(a)={\frac {1}{2\pi i}}\int _{C}{\frac {f(z)}{z-a}}dz.}$

## Cauchy integral formulas

The Cauchy Integral formulas for a complex valued function ${\displaystyle f(z)}$ which is analytic inside and on a simple closed curve ${\displaystyle C}$ in some region ${\displaystyle {\mathcal {R}}}$ of the complex ${\displaystyle z}$ plane, for a complex number ${\displaystyle a}$ inside ${\displaystyle C}$

${\displaystyle f^{(n)}(a)={\frac {n!}{2\pi i}}\int _{C}{\frac {f(z)}{(z-a)^{n+1}}}dz.}$