Cauchy Integral Formula

When $f$ is analytic inside and on a simple, closed, piecewise smooth curve $C$, its value at any point $z$ interior to $C$ is given by the contour integral

$$f(z)=\frac{1}{2\pi i}{\oint }_c\frac{f(\zeta )}{\zeta -z}d\zeta .$$

where the contour integral is taken in the counter-clockwise direction.

A couple of other variations on this, just by rearranging with algebra, and using $z_0$ instead of $z$, and $z$ instead of $\zeta$.

$${\oint }_C\frac{f(z)}{z-z_0}dz=2\pi if(z_0),f(z_0)=\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{z-z_0}dz.$$

Another useful formula that results from the Cauchy Integral formula:

When $f$ is analytic inside and on a closed, piecewise smooth curve $C$, and $n$ is a nonnnegative integer, the value of the contour integral

$${\oint }_C\frac{f(z)}{(z-z_0{)}^{n+1}}dz=\{\begin{array}{l}0,z_0\text{outside}C\hfill \\ \frac{2\pi i}{n!}{f}^{(n)}(z_0)z_0\text{inside}C.\hfill \end{array}$$