Cauchy's integral theorem

In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if ${\displaystyle f(z)}$ is holomorphic in a simply connected domain ⁠${\displaystyle \Omega }$⁠, then for any simply closed contour ${\displaystyle C}$ in ⁠${\displaystyle \Omega }$⁠, that contour integral is zero. $${\displaystyle \int _{C}f(z)\,dz=0.}$$