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Contour integrals and the residue theorem

Cauchy's theorem, the integral formula, and evaluating real integrals with residues.

The integral of a holomorphic function around a closed loop is zero; around a singularity it is 2πi times the residue. Real integrals like ∫dx/(1 + x²) close in the upper half-plane and pick up the residue at i. Picture it: the contour along the real axis closed by a great semicircle enclosing the pole at i. Think it: Cauchy's theorem is Green's theorem plus the Cauchy–Riemann equations.

דוגמה עובדת: integrate 1/(1+x^2) dx from -oo to oo

Calculus · Definite-Integrals

Integrate 1/(x^2 + 1) from -oo to oo

\int_{-\infty}^{\infty} \frac{1}{x^{2} + 1}\, dx

צעד אחר צעד

  1. \int_{-\infty}^{\infty} \frac{1}{x^{2} + 1}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int \frac{1}{x^{2} + 1}\, dx = \operatorname{atan}{\left(x \right)}

    Recognise an inverse-trig / inverse-hyperbolic form.

  3. F(\infty) - F(-\infty) = \left(\frac{\pi}{2}\right) - \left(- \frac{\pi}{2}\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \pi \approx 3.1416

    Simplify.

גלה את התשובה
\pi

Symbols used here

\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\arcsin,\ \sin^{-1}
inverse sine
The angle whose sine is the given value (and likewise arccos, arctan).
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
i
imaginary unit
i² = −1.
\approx
approximately equal
Equal to the precision shown, not exactly.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\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.

How to: Contour integrals and the residue theorem

  1. First find an antiderivative F, then evaluate F(b) − F(a).
  2. Recognise an inverse-trig / inverse-hyperbolic form.
  3. Fundamental theorem of calculus: plug in the limits.
  4. Simplify.

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.

נסה את שלך.

יותר בפנים. Complex Analysis

The complex plane and Euler's formulaHolomorphic functions and the Cauchy–Riemann equationsPower series and analytic continuation