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A complex function is one which maps a complex number x + iy into a new complex number u(x,y) + iv(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.

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