A **complex function** is one which maps a complex number *x +* i*y* into a new complex number *u(x,y) +* i*v(x,y)*. The field of complex analysis largely focuses on complex functions which are holomorphic, or complex differentiable.

## Operations of complex functions

### Differentiation

Given a complex function *f(z)*, differentiation is identical to real differentiation given that the function is differentiable; that is, if it respects the Cauchy–Riemann conditions.

### Integration

If a complex function is holomorphic, by Cauchy's theorem, a definite integral can be found by taking the antiderivative of its endpoints.

### Transcendental functions

Transcendental functions applied on complex numbers can be approximated by using a Taylor series. For example, to find *e* raised to the power of a complex number *z*,

- $ e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} $

The same can be done for trigonometric and hyperbolic functions.