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Line 1: − + − − : <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.
Removed wrong and irrelevent material
Complex analysis is the study of [[complex number]]s of the form:
Complex analysis is the study of functions supported on the [[complex number]s.
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.
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.