Evaluating Integrals by Residues

The residue theorem turns definite integrals — including real ones with no elementary antiderivative — into finite sums of residues. This is the single most-used technique from complex analysis in physics: every one-loop Feynman integral, every propagator, is evaluated this way. This page collects the standard contours and the physically crucial $i\epsilon$ prescription. It applies laurent-residues.md.

The basic strategy

To evaluate a real integral ${\int }_{-\infty }^{\infty }f(x)dx$: promote $x$ to a complex variable, close the contour with a large arc (in the upper or lower half-plane), show the arc's contribution vanishes, and equate the remaining real-axis integral to $2\pi i$ times the enclosed residues. The art is choosing the contour so the arc dies and the wanted integral survives.

Rational functions

For ${\int }_{-\infty }^{\infty }\frac{P(x)}{Q(x)}dx$ with $\mathrm{deg}Q\ge \mathrm{deg}P+2$ and no real poles, close with a semicircle of radius $R\to \infty$ in the upper half- plane. The arc scales as $R\cdot R^{\mathrm{deg}P-\mathrm{deg}Q}\to 0$, so $${\int }_{-\infty }^{\infty }\frac{P}{Q}dx=2\pi i\sum _{\text{UHP poles}}\mathrm{Res}\frac{P}{Q}.$$ Example: ${\int }_{-\infty }^{\infty }\frac{dx}{1+x^2}=2\pi i{\mathrm{Res}}_{z=i}\frac{1}{1+z^2}=2\pi i\cdot \frac{1}{2i}=\pi .$

Oscillatory integrals and Jordan's lemma

For Fourier-type integrals $\int f(x)e^{ikx}dx$ ($k>0$), the exponential $e^{ikz}=e^{ik(x+iy)}$ decays as $e^{-ky}$ in the upper half-plane, which controls the arc even when $f$ decays only like $1/R$:

Jordan's lemma. If $f(z)\to 0$ uniformly on the upper semicircle as $R\to \infty$ and $k>0$, then ${\int }_{\text{arc}}f(z)e^{ikz}dz\to 0$.

So ${\int }_{-\infty }^{\infty }f(x)e^{ikx}dx=2\pi i{\sum }_{\text{UHP}}\mathrm{Res}(f{e}^{ikz})$ for $k>0$ (close below for $k<0$). This is exactly how one Fourier- transforms propagators between momentum and position space, and the sign of $k$ choosing the half-plane is the seed of causality.

Poles on the contour and principal values

When a pole sits on the real axis, the integral is ambiguous until a prescription is chosen. Indenting the contour with a small semicircle of radius $ϵ$ around the pole contributes $\pm i\pi \mathrm{Res}$ (half a residue, sign by orientation), giving the Sokhotski–Plemelj formula $$\frac{1}{x-x_0\mp i\epsilon }=\mathrm{P}\frac{1}{x-x_0}\pm i\pi \delta (x-x_0),$$ where $\mathrm{P}$ is the Cauchy principal value. The real part is the principal value; the imaginary part is a delta function — the mathematical origin of the relation between dispersive and absorptive parts of amplitudes.

The $i\epsilon$ prescription

The most important physical instance: the Feynman propagator $$\frac{1}{p^2-m^2+i\epsilon }$$ displaces the on-shell poles $p^0=\pm (E_p-i\epsilon )$ off the real $p^0$-axis. The tiny $i\epsilon$ decides which way the $p^0$ contour passes each pole, and hence which frequencies propagate forward vs. backward in time — encoding causality (Feynman boundary conditions: positive energy forward, negative energy backward). Different prescriptions (retarded, advanced, Feynman) are different contours around the same poles. This is the entry point to loop integrals in interactions/feynman-rules.md and the analytic structure read off in observables/README.md.

Summing series by residues

The residue theorem evaluates infinite sums, not just integrals. The functions $\pi \cot (\pi z)$ and $\pi \csc (\pi z)$ have simple poles at every integer, with residues $1$ and $(-1{)}^n$ respectively; integrating $f(z)$ times one of them over a large square and letting it grow (the boundary contribution vanishing) equates the sum over integers to minus the sum of residues at the poles of $f$: $$\sum _{n=-\infty }^{\infty }f(n)=-\sum _{\text{poles of }f}\mathrm{Res}[\pi \cot (\pi z)f(z)],\sum _{n=-\infty }^{\infty }(-1{)}^{n}f(n)=-\sum _{\text{poles of }f}\mathrm{Res}[\pi \csc (\pi z)f(z)].$$ With $f(z)=1/z^2$ (a double pole at $0$, residue of $\pi \cot (\pi z)/z^2$ equal to $-{\pi }^2/3$) this gives the Basel sum ${\sum }_{n\ge 1}1/n^2={\pi }^2/6$. The same $\pi \cot$ kernel is the Mittag-Leffler expansion of the cotangent, and the underlying integer-to-frequency duality is Poisson summation — the bridge to the theta and zeta material.

Other standard tricks

• Trigonometric integrals ${\int }_0^{2\pi }R(\cos \theta ,\sin \theta )d\theta$: substitute $z=e^{i\theta }$, turning the integral into a contour integral around the unit circle.
• Keyhole contours for ${\int }_0^{\infty }{x}^{\alpha -1}R(x)dx$: wrap a branch cut along the positive axis (see riemann-surfaces.md); the discontinuity across the cut isolates the integral.
• Rectangular / wedge contours exploiting periodicity or a symmetry of the integrand (e.g. $\int e^{-x^2}\cos (bx)dx$).

References

• Arfken, Weber & Harris, Mathematical Methods for Physicists, Ch. 11.
• Stein & Shakarchi, Complex Analysis, Ch. 3.
• Peskin & Schroeder, An Introduction to QFT, §7 (the $i\epsilon$ prescription in loop integrals).