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Complex Analysis

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.

उपक्रम

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)

Symbols used here

|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\theta
theta
The usual name for an angle.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\iint,\ \oint
double / contour integral
Integral over a region of the plane; integral around a closed curve.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\zeta(s)
Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.
\operatorname{Res}_{z=a} f
residue
Coefficient of 1/(z − a) at a singularity; drives contour integrals.
f: \mathbb{C} \to \mathbb{C},\ \bar{z}
complex function, conjugate
A map of the complex plane; x − iy.

Questions people ask

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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