A holomorphic function is a complex function that is complex differentiable on every point in its domain; that is, the derivative exists and the power series converges. A complex function f(z) will be holomorphic iff
- The function is differentiable (that is, the partial derivatives exist and are continuous)
- The function preserves angles of intersection between curves (such a function is known as a conformal mapping)
- The determinant of the Jacobian matrix is greater than zero (if all other conditions are met but the determinant is less than zero, the function is called or anti-holomorphic or anti-conformal and will be differentiable with respect to the complex conjugate of z).
If a function is holomorphic it will obey the Cauchy–Riemann conditions (and by extension, Laplace's equation). If a function obeys the Cauchy-Riemann conditions and the derivative is not zero, it will also preserve angles, and so will be holomorphic if the other two conditions are met. All holomorphic functions are also analytic.
A function is called meromorphic if it is holomorphic on an open subset except for a finite number of points called poles. Any meromorphic function can be expressed as a fraction of two holomorphic functions, with the poles corresponding to zeros in the denominator. The gamma function is an example of a meromorphic function.