Cauchy's integral theorem

Theorem in complex analysis

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 f ( z) {\displaystyle f(z)} is holomorphic in a simply connected domain ⁠ Ω {\displaystyle \Omega } ⁠, then for any simple closed contour C {\displaystyle C} in ⁠ Ω {\displaystyle \Omega } ⁠, that contour integral is zero. ∫ C f ( z) d z = 0.

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