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Complex Analysis
Holomorphic functions, contour integrals, residues, the rigidity of the complex plane.
មេរៀន
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.
មេរៀនបន្ថែម
The complex plane
Holomorphic functions
Integration
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សំណួរដែលមនុស្សសួរ
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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