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 $C$

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