# Holomorphic Function > [!NOTE] Holomorphic Function > A [[Complex Number|complex-valued]] function is *holomorphic* on a region $R$ if at every point of $R$ there exists a [[neighbourhood]] in which the function is complex [[Derivative|differentiable]]. The requirement for holomorphic functions to be complex differentiable within a neighbourhood of all points of a region means that they are *infinitely differentiable*. It also means they are always locally equal to its own [[Taylor series]] and are thus also complex [[Analytic Function|analytic]] functions. As holomorphic functions and complex analytic functions are equivalent, it is common for holomorphic functions to simply be called analytic functions, despite analytic functions being a broader category that contains real functions as well. As holomorphic functions are complex differentiable everywhere, they also satisfy the [[Cauchy-Riemann equations]]. The real and imaginary parts of a holomorphic function are [[Harmonic Function|harmonic]] functions. ## Properties ### Combinations of holomorphic functions If two functions $f$ and $g$ in a domain $D$ are holomorphic, then $f\pm g$, $fg$, and $f\circ g$ are all *also holomorphic* as complex differentiation is linear and obeys the same differentiation rules as real differentiation. The quotient of two functions $f/g$ is only holomorphic if $g$ has *no zeroes* in $D$. ### Cauchy's integral theorem [[Cauchy's integral theorem]] states that the [[contour integral]] of any holomorphic function $f(z)$ along a simple closed contour $\gamma$ in a [[Simply Connected Space|simply connected]] domain is *always zero*. $\int_{\gamma}f(z)\;dz=0$ ### Cauchy's integral formula [[Cauchy's integral formula]] relates the values of a function $f(z)$ that is holomorphic on a disk to the values on the boundary $\gamma$ of the disk. $f(a)=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(z)}{z-a}\;dz$