komplexfunktion

A complex function is a function whose argument and/or value are complex numbers. Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i² = -1. A complex function f can be expressed as f(z) where z is a complex number. The output of the function, f(z), is also a complex number.

These functions can be thought of as mapping points in the complex plane to other points in the complex plane. For instance, a common complex function is f(z) = z², which squares a complex number. If z = x + yi, then z² = (x + yi)² = x² + 2xyi + (yi)² = x² - y² + 2xyi. This shows how the real and imaginary parts of the input are transformed to produce the real and imaginary parts of the output.

The study of complex functions is a major branch of mathematics known as complex analysis. This field explores properties such as differentiability (analyticity), integration, and series expansions of complex functions. Analytic functions are particularly important as they possess a rich set of properties that differ significantly from real-valued functions. Concepts like Cauchy-Riemann equations are fundamental for determining if a complex function is analytic. Complex analysis has wide-ranging applications in physics, engineering, and other scientific disciplines.