Complex analysis is the branch ofmathematics investigatingholomorphic functions, i.e. functions which are defined in some region of thecomplex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real)differentiability. For instance, every holomorphic function is representable aspower series in every open disc in its domain of definition, and is thereforeanalytic. In particular, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, such as allpolynomials, theexponential function, and thetrigonometric functions, are holomorphic.See also :holomorphic sheaves andvector bundles.
