Contour Integration

Introduction

In real analysis, we integrate functions along the real line, moving from one point to another along a straight path. Complex analysis opens a richer world: we can integrate functions along curves in the complex plane, winding through two-dimensional space in any manner we choose. This freedom to choose different paths—and the remarkable theorems governing how the choice of path affects the integral—lies at the heart of contour integration.

Contour integration extends the familiar concept of integration to complex-valued functions of a complex variable. Given a function f(z) and a curve γ in the complex plane, we compute the integral of f along γ. The result depends, in general, on both the function and the path taken.

For analytic functions, however, the integral often depends only on the endpoints—or vanishes entirely for closed paths. Cauchy’s theorem and the residue theorem transform difficult real integrals into tractable complex ones and provide powerful tools for evaluating sums, solving differential equations, and understanding global behavior of analytic functions.

Curves in the Complex Plane

A curve in the complex plane is a continuous function from a real interval into C.

Parametrized Curves

A parametrized curve is a continuous function γ:[a,b]→C.

We write γ(t)=x(t)+i*y(t), where x(t) and y(t) are real-valued functions. The point γ(a) is the initial point and γ(b) the terminal point. If γ(a)=γ(b), the curve is closed.

Smooth and Piecewise Smooth Curves

A curve γ is smooth if γ(t)′ exists, is continuous, and never vanishes on (a,b).

A curve is piecewise smooth if there exist points a=(t_0)<(t_1)<⋯<(t_n)=b such that γ is smooth on each subinterval [(t_k−1),(t_k)].

Contours

A contour is a piecewise smooth curve used as a path of integration. Orientation matters. If γ runs from (z_0) to (z_1), then −γ denotes the same path traversed in reverse.

Definition of the Contour Integral

Let f be continuous on a contour γ:[a,b]→C. The contour integral is defined by (∫_γ^)(f(z)*d(z))=(∫_a^b)(f*(γ(t))γ(t)′*d(t)).

Writing f(z)=u(x,y)+i*v(x,y) and d(z)=d(x)+i*d(y), we obtain (∫_γ^)(f(z)*d(z))=(∫_γ^)((u*d(x)−v*d(y))+i*(∫_γ^)((v*d(x)+u*d(y)))).

Thus a complex contour integral equals two real line integrals.

Independence of Parametrization

If two parametrizations trace the same curve in the same direction, the integrals agree:

(∫_γ^)(f(z)*d(z))=(∫_σ^)(f(z)*d(z)).

Properties of Contour Integrals

Linearity

(∫_γ^)((α*f(z)+β*g(z))*d(z))=α*(∫_γ^)(f(z)*d(z))+β*(∫_γ^)(g(z)*d(z)).

Additivity

If γ=(γ_1)+(γ_2), (∫_γ^)(f(z)*d(z))=(∫_(γ_1)^)(f(z)*d(z))+(∫_(γ_2)^)(f(z)*d(z)).

Reversal

(∫_−γ^)(f(z)*d(z))=−(∫_γ^)(f(z)*d(z)).

ML Inequality

If |f(z)|≤M on γ and L is its length,|(∫_γ^)(f(z)*d(z))|≤M*L,L=(∫_a^b)(|γ(t)′|d(t)).

Fundamental Theorems

Cauchy’s Theorem

If f is analytic on and inside a simple closed contour γ, (∮_γ)(f(z)*d(z))=0.

If f is analytic on a simply connected domain D, then (∮_γ)(f(z)*d(z))=0 for every closed contour γ⊂D.

Path Independence

If f is analytic on a simply connected domain, (∫_(γ_1)^)(f(z)*d(z))=(∫_(γ_2)^)(f(z)*d(z)).

An antiderivative can be defined by F(z)=(∫_(z_0)^z)(f(ζ)*d(ζ)),F(z)′=f(z), and (∫_γ^)(f(z)*d(z))=F((z_1))−F((z_0)).

Cauchy’s Integral Formula

If f is analytic inside and on γ and (z_0) lies inside, f((z_0))=1/(2*π*i)*(∮_γ)(f(z)/(z−(z_0))*d(z)).

For derivatives, f((z_0))(n)=(n!)/(2*π*i)*(∮_γ)(f(z)/((z−(z_0))(n+1))*d(z)).

Residue Theorem

If f is analytic except for isolated singularities (z_k) inside γ, (∮_γ)(f(z)*d(z))=2*π*i*(∑_^)(Res(f,(z_k))).

For a simple pole, Res(f,(z_0))=(lim_)(z−(z_0))*f(z).

For a pole of order n, Res(f,(z_0))=1/((n−1)!)*(lim_)((d()(n−1))/(d*z(n−1)))*[(z−(z_0))n*f(z)].

Applications

Contour integration evaluates difficult real integrals, sums infinite series, computes inverse Laplace and Fourier transforms, and analyzes differential equations.

Contour Deformation

If two contours can be continuously deformed into each other without crossing singularities,

(∫_(γ_1)^)(f(z)*d(z))=(∫_(γ_2)^)(f(z)*d(z)).

Choosing contours cleverly—avoiding singularities and exploiting symmetry—is central to effective contour integration.