Complex analysis
Complex analysis is the study of complex numbers of the form:
- <math>i = \sqrt{-1}</math>
From this definition a "complex plane" is constructed, consisting of z = x + iy, where x and y are real numbers:
- <math>z = x + iy\,</math>, and
- <math>w = f(z) = u(z) + iv(z)\,</math>
- where <math>x,y \in \mathbb{R}\,</math> and <math>u(z), v(z)\,</math> are real-valued functions.
so that real numbers are on the x-axis and imaginary numbers are on the y-axis.
Traditionally, mathematical objects are derived from physics or geometry. The complex numbers on the other hand were constructed out of the blue as an ad-hoc solution to certain problems that appeared to be unsolvable with real numbers. Initially unanimously rejected by the leading mathematicians, they were eventually accepted by a growing number of mathematicians due to a series of spectacular results. Not all of those results can be proved with elementary techniques however, and are thus considered questionable by more rigorous mathematicians.
Much of complex analysis is devoted to studying holomorphic functions that are infinitely differentiable. These functions take complex values in the complex plane and are differentiable as complex functions.
Complex analysis relies heavily on contour integration, which enables computation of difficult integrals by examining singularities of the function in regions of the complex plane near the limits of integration.
The central result in complex analysis is the Cauchy integral theorem, and a powerful claim of complex analysis is Picard's great theorem.
The Cauchy-Riemann equations provide conditions a function must satisfy in order for a complex generalization of the derivative (the "complex derivative"). When the complex derivative can be defined "everywhere," the function is called "analytic".
Additional concepts in complex analysis include the following: