Essential Mathematics for QFT

4. Complex Analysis & Contour Integration

4.1 Analytic functions

Write $z = x + iy$ and $f(z) = u(x,y) + i\,v(x,y)$. The derivative is defined as usual, but with a far stronger requirement: the limit must come out the same from every direction in the plane.

Approaching along the real axis ($\Delta z = \Delta x$) gives $f' = \partial_x u + i\,\partial_x v$; approaching along the imaginary axis ($\Delta z = i\Delta y$) gives $f' = -i\,\partial_y u + \partial_y v$. Equating real and imaginary parts gives the Cauchy–Riemann equations:

A function differentiable throughout a neighbourhood is called analyticthere. The condition is so restrictive that analytic functions are automatically infinitely differentiable, and are determined everywhere by their values on any small patch — the fact that makes analytic continuation possible at all.

4.2 Cauchy's theorem

If $f$ is analytic everywhere inside and on a closed contour $C$, then

The quickest route is Green's theorem. Writing $f\,dz = (u+iv)(dx + i\,dy)$ and separating,

Both integrands vanish identically by Cauchy–Riemann. The practical consequence is the one used constantly below: a contour may be deformed freely through any region where the function is analytic, without changing the integral. Only singularities pin it down.

4.3 Laurent series and residues

Near an isolated singularity $z_0$ a function has a Laurent expansion, a Taylor series extended to negative powers:

The coefficient $a_{-1}$ is the residue. It earns a name because of a single elementary integral: on a small circle about $z_0$, substituting $z - z_0 = \rho e^{i\theta}$ gives

Every term integrates to zero except one. For a simple pole the residue is $\mathrm{Res} = \lim_{z\to z_0}(z-z_0)f(z)$; for a pole of order $m$,

4.4 The residue theorem

Now the two previous subsections combine, and the derivation is three sentences. Deform $C$ into small circles around each enclosed singularity — legal by 4.2, since the function is analytic in between. On each circle expand in a Laurent series and integrate term by term. Only the $a_{-1}$ term survives:

That is the whole theorem. An integral around a closed loop does not care about the loop, only about which singularities it encircles — which is why it converts hard real integrals into arithmetic.

4.5 Worked example: a real integral by closing the contour

Take $\int_{-\infty}^{\infty} dx/(1+x^2)$, whose answer you already know from $\arctan$. Regard it as a complex integral and close the path with a large semicircle in the upper half-plane. On the arc of radius $R$ the integrand falls as $R^{-2}$ while the arc length grows as $R$, so that contribution vanishes as $R\to\infty$. The closed contour encircles only the pole at $z = i$, with residue $1/(2i)$:

The pattern generalises: choose a contour that contains the integral you want, make the rest vanish, and read off the residues. Which contour to choose is the only real skill.

4.6 A pole on the path — and what iε actually means

The case that matters most in physics is the awkward one: a pole sitting on the integration path itself, so the integral is not even defined. The cure is to indent the contour with a small semicircle of radius $\varepsilon$ and take $\varepsilon\to 0$. The straight parts give the Cauchy principal value; the semicircle gives half a residue, with a sign set by which way you go round:

This is the Sokhotski–Plemelj formula, and it is the entire content of the $i\epsilon$ prescription. Displacing a pole off the real axis is just a compact way of saying which side to indent — that is, which boundary condition you are imposing. In a propagator,

the choice of sign selects causal propagation: positive-energy solutions forward in time, negative-energy ones backward. It is not a regularisation trick, and nothing is being swept away — the two terms above are the principal value and a genuine delta function, and both have physical consequences.

4.7 Worked example: rotating the contour (the Fresnel integral)

Not every use involves a pole. Consider $\int_{-\infty}^{\infty} e^{iax^2}dx$ with $a>0$, which does not converge absolutely and cannot be done by any real substitution. The integrand is entire — no singularities at all — so by 4.2 the contour can be rotated freely. Put $z = e^{i\pi/4}u$, so that $iaz^2 = i a\, e^{i\pi/2} u^2 = -au^2$:

An oscillatory integral has become an ordinary Gaussian by turning the path through 45°. This single result is the origin of the factor $2\pi$ in the Landau–Zener formula, and of the phase conventions in every stationary-phase and path-integral calculation later in this course.

4.8 Analytic continuation and Wick rotation

Because an analytic function is fixed by its values on any patch, a function defined in one region has at most one analytic extension to a larger one. QFT exploits this constantly: rotating $t \to -i\tau$ turns the Minkowski metric into a Euclidean one, oscillatory path integrals into convergent ones, and $e^{iS}$ into $e^{-S_E}$.

The rotation is legitimate precisely when no singularity is crossed as the contour turns — which is the same deformation argument as 4.2, and the same one used in 4.7. The $i\epsilon$ prescription is what guarantees the poles sit in the two quadrants the rotation avoids.