Residue Theory¶

Job: evaluate integrals and count zeros/poles via residues; argument principle and Rouché as deformation tools; standard contour devices for real integrals and sums.

Core¶

Residues and the residue theorem¶

If $f$ has an isolated singularity at $a$, the residue is the Laurent coefficient $c_{-1}$: $$ \operatorname{Res}(f,a)=c_{-1}=\frac{1}{2\pi i}\int_{|z-a|=\varepsilon}f(z)\,dz. $$

Singularity	Residue formula
Simple pole	$\lim_{z\to a}(z-a)f(z)$
Simple pole, $p/q$	$p(a)/q'(a)$ if $q(a)=0$ simple, $p(a)\neq 0$
Pole order $k$	$\frac{1}{(k-1)!}\lim_{z\to a}\partial_z^{k-1}\bigl((z-a)^k f(z)\bigr)$
Removable	$0$
Laurent known	read off $c_{-1}$
At $\infty$	$\operatorname{Res}(f,\infty)=-c_{-1}$ of $f(z)\,dz$ in the chart $w=1/z$; equivalently $-\frac{1}{2\pi i}\int_{

Classification reminder. Removable / pole order $k$ / essential — by Laurent tail; Casorati–Weierstrass / Picard for essential singularities (analytic functions).

Theorem (Residue theorem). $f$ holomorphic in $U$ except isolated singularities $a_j$; $\gamma$ a cycle homologous to $0$ in $U$, avoiding the $a_j$: $$ \frac{1}{2\pi i}\int_\gamma f=\sum_j n(\gamma,a_j)\operatorname{Res}(f,a_j). $$ Global constraint. For $f$ meromorphic on $\widehat{\mathbb{C}}$ (e.g. rational), $\sum_{\text{all }a\in\widehat{\mathbb{C}}}\operatorname{Res}(f,a)=0$.

Argument principle and Rouché¶

Theorem (Argument principle). $f$ meromorphic inside and on a contour $\gamma$ (no zeros/poles on $\gamma$), $\gamma$ standard positive boundary: $$ \frac{1}{2\pi i}\int_\gamma\frac{f'}{f}\,dz=N-P $$ (zeros minus poles, counted with multiplicity). Equals $\frac{1}{2\pi}\Delta_\gamma\arg f$ (winding of $f\circ\gamma$ about $0$).

Variants. $\frac{1}{2\pi i}\int_\gamma g\frac{f'}{f}$ sums $g$ at zeros minus poles; logarithmic derivatives convert product data to sums of residues.

Rouché. If $|g|<|f|$ on $\gamma$, then $f$ and $f+g$ have the same number of zeros inside $\gamma$ (counted with multiplicity). Proof: $H(t)=f+tg$ never vanishes on $\gamma$; $N(t)-P(t)$ continuous in $t\in[0,1]$ hence constant.

Corollary	Use
FTA	$z^n$ dominates lower terms on large $
Zero location	compare dominant terms on circles/annuli
Hurwitz	locally uniform limits of holomorphic $f_n\neq 0$ are zero-free or identically zero

Open mapping / maximum modulus sit upstream; argument principle upgrades them to quantitative zero counts.

Real integrals via contours¶

Integral type	Contour / device
$\int_{-\infty}^\infty R(x)\,dx$ rational, $\deg\mathrm{den}\ge\deg\mathrm{num}+2$	semicircle; sum residues in closed half-plane
$\int R(x)\sin x$, $\cos x$	$\operatorname{Im}/\operatorname{Re}$ of $e^{iz}R(z)$; Jordan’s lemma on vanishing arcs
$\int_0^{2\pi}R(\cos\theta,\sin\theta)\,d\theta$	$z=e^{i\theta}$, $d\theta=dz/(iz)$ on unit circle
Keyhole	$z^{a-1}R(z)$ or $\operatorname{Log}$; branch cut $[0,\infty)$
Rectangular / cotangent	sums via $\pi\cot(\pi z)$; strip contours for Fourier
Principal value + indents	small semicircles about real poles contribute $\pm i\pi\operatorname{Res}$ (orientation)

Decay hypotheses are mandatory: always estimate the arc; polynomial decay or exponential decay from $e^{iz}$ in the correct half-plane.

Summation formulas (pointer)¶

$\pi\cot(\pi z)$ has simple poles at $\mathbb{Z}$ with residue $1$. For suitable rational $f$ (decay at $\infty$, poles controlled): $$ \sum_{n\in\mathbb{Z}}f(n)=-\sum_j\operatorname{Res}\bigl(\pi\cot(\pi z)f(z)\bigr)\Big|_{\text{poles of }f}. $$ Analogues: $\pi/\sin(\pi z)$, $\pi/\cos$. Partial fraction expansions of trigonometric functions are residue calculus in disguise.

See it¶

Nyquist stability. The argument principle applied to a transfer/characteristic function along the imaginary axis (indented) counts RHP zeros — control theory packaging of $N-P$.

Explicit formulae / zero counting for $\zeta$. Contour integrals of $\zeta'/\zeta$ on rectangles estimate $N(T)$; the same logarithmic derivative appears in explicit formulae relating zeros to primes. Residue calculus is the skeleton of analytic number theory’s contour shifts.

Partial fractions of $\pi\cot$. Expanding and matching residues recovers $\sum 1/n^2=\pi^2/6$ and higher even zeta values — a prototype of evaluating Euler sums by residues.

Check¶

Compute $\int_{-\infty}^\infty\frac{dx}{1+x^2}$ by residues.

$f(z)=1/(1+z^2)$, simple pole at $i$ in the UHP, $\operatorname{Res}(f,i)=1/(2i)$. Arc contribution $\to 0$ as $R\to\infty$. Integral $=2\pi i\cdot\frac{1}{2i}=\pi$.

Use Rouché to show $z^5+3z+1$ has all five zeros in $|z|<2$.

On $|z|=2$: $|3z+1|\le 7$, $|z^5|=32>7$, so $z^5$ and $z^5+3z+1$ have the same number of zeros inside ($5$).

Explain why $\sum\operatorname{Res}(f)=0$ on $\widehat{\mathbb{C}}$ for rational $f\,dz$.

On a large circle, $\int_{|z|=R}f=2\pi i\sum_{\text{finite poles}}\operatorname{Res}$; as $R\to\infty$ the same integral equals $-2\pi i\operatorname{Res}(f,\infty)$ with orientation conventions, hence total sum vanishes. Alternatively: meromorphic $1$-forms on $\mathbb{P}^1$ have residue sum zero (degree of canonical / residue theorem on compact Riemann surfaces).

Pitfalls¶

Residue at $\infty$ uses the local coordinate $w=1/z$ and a sign from $dz=-dw/w^2$; do not merely read $c_{-1}$ of $f(z)$ in $z$ without transforming.
Order of pole drops when numerator vanishes; always cancel common factors before applying $p/q'$ formulas.
Branch cuts: residue theorem needs a single-valued holomorphic branch on the contour’s region; keyhole/indent contours encode the jump.
Jordan lemma vs ML inequality: exponential decay in $\operatorname{Im} z>0$ for $e^{iz}$ fails in the lower half-plane — choose half-plane by the sign of the exponential.
Multiplicities in $N-P$ are easy to drop when zeros sit on symmetries or when $f$ shares factors with $f'$.

Refs¶

Ahlfors, Complex Analysis
Conway, Functions of One Complex Variable
Stein–Shakarchi, Complex Analysis
Markushevich, Theory of Functions of a Complex Variable

Next¶

Complex integration · Analytic functions · Foundations · Complex analysis overview · Analytic number theory · Real analysis

Singularity	Residue formula
Simple pole	\(\lim_{z\to a}(z-a)f(z)\)
Simple pole, \(p/q\)	\(p(a)/q'(a)\) if \(q(a)=0\) simple, \(p(a)\neq 0\)
Pole order \(k\)	\(\frac{1}{(k-1)!}\lim_{z\to a}\partial_z^{k-1}\bigl((z-a)^k f(z)\bigr)\)
Removable	\(0\)
Laurent known	read off \(c_{-1}\)
At \(\infty\)	\(\operatorname{Res}(f,\infty)=-c_{-1}\) of \(f(z)\,dz\) in the chart \(w=1/z\); equivalently $-\frac{1}{2\pi i}\int_{

Integral type	Contour / device
\(\int_{-\infty}^\infty R(x)\,dx\) rational, \(\deg\mathrm{den}\ge\deg\mathrm{num}+2\)	semicircle; sum residues in closed half-plane
\(\int R(x)\sin x\), \(\cos x\)	\(\operatorname{Im}/\operatorname{Re}\) of \(e^{iz}R(z)\); Jordan’s lemma on vanishing arcs
\(\int_0^{2\pi}R(\cos\theta,\sin\theta)\,d\theta\)	\(z=e^{i\theta}\), \(d\theta=dz/(iz)\) on unit circle
Keyhole	\(z^{a-1}R(z)\) or \(\operatorname{Log}\); branch cut \([0,\infty)\)
Rectangular / cotangent	sums via \(\pi\cot(\pi z)\); strip contours for Fourier
Principal value + indents	small semicircles about real poles contribute \(\pm i\pi\operatorname{Res}\) (orientation)