Cauchy's integral formula

Cauchy's integral formula is a theorem in complex analysis which relates a holomorphic function defined on a disk to its values on the edge of the disk, and gives a formula for $f(a)$ for every $a$ within the disk, or outside if the function is holomorphic everywhere.

$f^{(n)}(a)={\dfrac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{n+1}}}dz$

If $n=0$ , this simplifies to

$f(a)={\dfrac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}dz$

See also

• Cauchy's integral theorem
• Green's theorem