# Cauchy's Integral Theorem **The [[contour integral]] of any function that is [[Holomorphic Function|holomorphic]] on a [[Simply Connected Space|simply connected]] domain over a simple closed contour on the same domain is always zero.** ![[Cauchy'sIntegralTheorem.svg]] *The contour integral of the function is always equal to zero regardless of the size, position, and shape of a simple closed contour containing regions where the function is holomorphic* > [!NOTE] Cauchy's Integral Theorem > Suppose $D\subseteq\mathbb{C}$ is a simply connected domain and $f:D\rightarrow\mathbb{C}$ is a holomorphic function. > > *Cauchy's integral theorem*, also known as the *Cauchy-Gorsat theorem*, states that for any simple closed contour $\gamma:[a,b]\rightarrow D$, > $\int_{\gamma}f(z)\;dz=0$ Cauchy's integral theorem implies that given a holomorphic function over a simply connected domain, the shape of any contour in that domain is irrelevant to the value of the contour integral which is always zero. ## Homotopy version The theorem can also be extended to the contour integrals of two [[Homotopy|homotopic]] [[Contour Integral#^39e92f|positive]] simple closed contours. > [!NOTE] Homotopy version of Cauchy's integral theorem > Suppose $\gamma_{1}$ and $\gamma_{2}$ are both positive simple closed contours such that $\gamma_{2}$ is in the interior[^1] of $\gamma_{1}$. If $f$ is a holomorphic function on and between the contours, then > $\int_{\gamma_{1}}f(z)\;dz=\int_{\gamma_{2}}f(z)\;dz$ The homotopy version of the theorem implies that any simple closed contour can be *continuously transformed* into *any other* without affecting the value of the integral, as long as they are both defined within a region where the integrand is holomorphic and the transformation remains within this region. ![[Cauchy'sIntegralTheoremHomotopy.svg|450]] *As the function is holomorphic on and between $\gamma_{1}$ and $\gamma_{3}$, the contour integral of any contour $\gamma_{2}$ in the region between them (light blue) will have the same value as on them* [^1]: This is not the same interior as the [[interior]] of a set, but is the region which is bounded by $\gamma_{1}$.