Deformation principle

The deformation principle: if $f$ is analytic on a domain $\Omega$ (possibly with holes, as long as $f$ stays analytic everywhere on $\Omega$), and a closed contour $C$ is continuously deformed to another closed contour $C^{\prime }$ entirely within $\Omega$, then

${\oint }_{C}f(z)dz={\oint }_{C^{\prime }}f(z)dz.$

So closed-contour integrals of analytic functions don’t change under continuous deformation through the analytic region.

Why this is true

The annular region between $C$ and $C^{\prime }$ is bounded by $C$ (counterclockwise) and $C^{\prime }$ (clockwise, or equivalently $-C^{\prime }$ counterclockwise). Connect them with thin corridors traversed in both directions. The composite contour bounds a region where $f$ is analytic, so by Cauchy’s theorem applied to that region:

${\oint }_{C}fdz-{\oint }_{C^{\prime }}fdz=0,$

since the corridor contributions cancel pairwise. Rearranging gives the deformation principle.

How this is used

Shrinking around a singularity. If $C$ encloses an isolated singularity $z_0$ of $f$, deform $C$ to a small circle of radius $\epsilon$ around $z_0$. On the small circle you can compute the integral by direct parameterization. The result holds for any $C$ enclosing $z_0$, whatever its shape and size.

Example. $\oint dz/(z-z_0)$ for any closed contour enclosing $z_0$ once. Deform to $|z-z_0|=\epsilon$, parameterize $z=z_0+\epsilon e^{it}$, integrate: get $2\pi i$. Holds for any contour by deformation.

Splitting around multiple singularities. If $C$ encloses singularities $z_1,\dots ,z_k$, split $C$ into the sum of $k$ contours, each enclosing one singularity, plus a contour with no singularities inside (which contributes zero by Cauchy’s theorem). Combined with the Cauchy integral formula or Residue theorem, this reduces complicated contour integrals to a sum of simple ones.

Deforming to large semicircles. For real improper integrals ${\int }_{-\infty }^{\infty }f(x)dx$ with $f$ extending to a rational function decaying at infinity: close the real-axis contour with a large semicircle in the upper half-plane. The semicircle contribution vanishes by the ML estimate (or Jordan’s lemma for $f(x)\cos (ax)$-type integrands). The original real integral equals the closed-contour integral, which equals $2\pi i$ times sum of residues in the upper half-plane.

Vector-calculus parallel

Same idea from vector calculus. The 2D rotational field $\mathbf{F}=⟨-y,x⟩/(x^2+y^2)$ has zero curl away from origin; on an annulus between two closed curves both enclosing the origin, Green’s theorem gives zero net contribution from the annulus, so the circulation is the same around either curve. Line integrals move freely between curves enclosing the same set of singularities.

This is the deformation principle of 2D vector calculus, the same thing as the complex-analytic version.

In context

The deformation principle does most of the work in contour integration. Combined with:

• Cauchy’s theorem (no singularities ⇒ integral is zero),
• Cauchy integral formula (one singularity of the form $g(z)/(z-z_0)$ ⇒ integral is $2\pi ig(z_0)$),
• Residue theorem (any isolated singularities ⇒ integral is $2\pi i\sum \mathrm{Res}$),

every closed-contour integral collapses to algebra at the singularities.