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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. Line 24:
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→top: re-categorisation
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.
*[[Argument Principle]]
*[[Argument Principle]]
*[[Branch Cut]] and [[Branch Point]]
*[[Branch Cut]] and [[Branch Point]]
*[[Residue theory]]
*[[Residue theory]] and [[Cauchy's residue theorem]]
*[[Conformal transformation]]
*[[Conformal transformation]]
*[[Contour Integration]]
*[[Contour Integration]]
*[[Euler's identity]]
*[[Euler's formula]]
*[[Laurent Series]]
*[[Laurent Series]]
*[[Morera's Theorem]]
*[[Morera's Theorem]]
*[[Elliptic function]]s
*[[Elliptic function]]s
[[category:mathematics]]
[[Category:Mathematics]]
[[Category:Complex Analysis]]