Cauchy's integral theorem

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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 Ω, then for any simply closed contour ${\displaystyle C}$ in Ω, that contour integral is zero.

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