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Complex Analysis

Holomorphic functions, contour integrals, residues, the rigidity of the complex plane.

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Introductory The complex plane and Euler's formula z = x + iy as a point, modulus and argument, e^{iθ} = cos θ + i sin θ. e^(i*pi) Core Holomorphic functions and the Cauchy-Riemann equations What complex differentiability demands, and why it is so restrictive. derivative of x^2 - y^2 wrt x Advanced Contour integrals and the residue theorem Cauchy's theorem, the integral formula, and evaluating real integrals with residues. integrate 1/(1+x^2) dx from -oo to oo Advanced Power series and analytic continuation Radius of convergence, the identity theorem, and extending ζ(s) beyond its series. taylor series of 1/(1-x)

Calculus over the complex numbers turns out to be astonishingly rigid: once a function is differentiable on a disc, it is infinitely differentiable, equals its Taylor series, and is determined everywhere by its values on a tiny set. The residue theorem then evaluates real integrals no real method can.

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The complex plane

Complex analysisComplex numberComplex planeEuler's formulaArgument (complex analysis)Complex conjugateRiemann sphereMöbius transformation

Holomorphic functions

Holomorphic functionCauchy–Riemann equationsAnalytic functionHarmonic functionConformal mapEntire functionMeromorphic functionZeros and polesBranch point

Integration

Cauchy's integral theoremCauchy's integral formulaResidue theoremLaurent seriesContour integrationArgument principleLiouville's theorem (complex analysis)Maximum modulus principleAnalytic continuationRiemann mapping theoremRiemann surface

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Why is complex differentiability so much stronger than real?

The limit must be the same from every direction in the plane, not just two. That forces the Cauchy-Riemann equations, which in turn force infinitely many derivatives and a convergent Taylor series.

What is a residue?

The coefficient of 1/(z − a) in the Laurent series at a singularity a. The residue theorem says a contour integral equals 2πi times the sum of the residues inside, which evaluates many real integrals in one line.

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