Complex analysis

Complex analysis is the study of complex numbers of the form:

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From this definition a "complex plane" is constructed, consisting of z = x + iy, where x and y are real numbers:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x + iy\,} , and

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where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y \in \mathbb{R}\,} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(z), v(z)\,} are real-valued functions.

so that real numbers are on the x-axis and imaginary numbers are on the y-axis.

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:

• Analytic Continuation
• Argument Principle
• Branch Cut and Branch Point
• Residue theory
• Conformal transformation
• Contour Integration
• Euler's identity
• Laurent Series
• Morera's Theorem
• Polygenic Function
• Elliptic functions