Complex analysis
![Plot of the function
f(x) = (x2 − 1)(x − 2 − i)2
/ (x2 + 2 + 2i). The hue represents the function argument, while the brightness represents the magnitude.](/Images/godic/202502/16/Color_complex_plot4407.jpg")
![The Mandelbrot set, a fractal.](/Images/godic/202502/16/Mandel_zoom_00_mandelbrot_set4407.jpg")
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.