Complex Analysis, Algebra & Topology

Singularities and Residues

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1. Laurent series, classification of singularities, Casorati–Weierstrass

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Lecture 9.1 - Singularities of a holomorphic function

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Lecture 9.2 - Pole of a function

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

9.3 - Laurent series

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

The expansion depends on the annulus — a standard exam trap.

Lecture - 9.4 Casorati Weierstrass theorem

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Singularities — three types, three tests free

In plain words

Count the negative powers. None means the trouble was fake, finitely many means a pole, and infinitely many means essential. In practice you take a limit rather than expanding anything.

The trichotomy at an isolated singularity $z_{0}$

Type	Laurent principal part	Behaviour of \|f\| near $z_{0}$	$\lim (z - z_{0})f$
Removable	none	bounded	0
Pole of order m	finitely many terms, up to $(z - z_{0})^{-m}$	$\to \infty$	finite non-zero for m = 1
Essential	infinitely many terms	wildly oscillating	no limit

Riemann's removable singularity theorem: bounded near $z_{0} \Rightarrow$ removable. Even $(z - z_{0})f(z) \to 0$ suffices.
Casorati–Weierstrass: near an essential singularity the image of every punctured neighbourhood is dense in $\mathbb{C}$ .
Picard (great): it in fact hits every value, with at most one exception, infinitely often.

Standard examples

f	at 0
sin z / z	removable (limit 1)
$(1 - \cos z)/z^{2}$	removable (limit ½)
$1/z^{3}$	pole of order 3
$e^{1/z}$	essential
sin(1/z)	essential
z $e^{1/z}$	essential (not a pole, despite the factor z)
$e^{z + 1/z}$	essential; residue $\sum 1/(k!(k+1)!)$
$\log z, \sqrt{z}$	not isolated — branch point, no Laurent series

Computing residues

simple pole: Res $= \lim (z - z_{0})f(z)$ ; for f = p/q with q simple zero: $p(z_{0})/q'(z_{0})$ .
pole of order m: Res = (1/(m−1)!) lim $d^{m-1}/dz^{m-1}$ $[(z - z_{0})^m f]$ .
essential: read a₋ $_{1}$ off the Laurent series (multiply the two series and collect $z^{-1})$ .
at $\infty$ : Res $_\infty f = -$ Res $_{0} (1/z^{2})f(1/z)$ ; the sum of all residues including $\infty$ is 0.

Laurent expansion depends on the annulus

1/((z−1)(z−2)) has three different Laurent expansions: on |z| < 1, on 1 < |z| < 2, and on |z| > 2. Exam questions always specify the annulus — read it.

Key takeaways

Bounded ⇒ removable is the cheapest test; |f| $\to \infty \Rightarrow$ pole.
$e^{1/z}-type$ factors always give essential, never poles.
The same function has different Laurent series on different annuli.

Classifying an isolated singularity in one lookinteractive

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Removable, pole or essential — decided by the Laurent tail, with the test for each.

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Exam focus: Classify a singularity from the Laurent tail or from the behaviour of |f| nearby: bounded ⇒ removable, → ∞ ⇒ pole, neither ⇒ essential (and then the image of every punctured neighbourhood is dense).

Practice

Part B2025 Decembershow ▾

Let h be a holomorphic function on

\mathbb{C}

{0} such that

lim_{|z|\to\infty}

h(z) = 0. For every n ≥ 1, consider

f_{n}(z) = \sum_{k=1}^n

h(zᵏ). Let 𝔻 denote the open unit disc {

z \in \mathbb{C}

: |z| < 1}. Which of the following statements is necessarily true?

A.For all z with |z| ≥ 1, the sequence { $f_{n}(z)$ } $_{n}\ge1$ converges.
B.For all z ∈ 𝔻{0}, the sequence { $f_{n}(z)$ } $_{n}\ge1$ converges.
C.The sequence { $f_{n}$ } $_{n}\ge1$ converges pointwise to a holomorphic function on 𝔻[0,1).
D.The sequence { $f_{n}$ } $_{n}\ge1$ converges pointwise to a holomorphic function on { $z \in \mathbb{C}$ : |z| > 1}.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

For

z \in \mathbb{C}

\ {0}, let f(z) = (1/z)sin(1/z) and g(z) = f(z)sin(z). Which of the following statements are true?

A.f has an essential singularity at 0✓ Correct
B.g has an essential singularity at 0✓ Correct
C.f has a removable singularity at 0
D.g has a removable singularity at 0

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let disc 𝔻 = {

z \in \mathbb{C}

: |z| < 1} and f be a holomorphic function on 𝔻 such that the function g(z) = e^(1/z)f(z) on 𝔻 \ {0} is bounded. Which of the following statements are true?

A.f(0) = 0✓ Correct
B.f(z) = 0 for all z ∈ 𝔻.✓ Correct
C.There exists a nonzero constant c such that f(z) = ce^(−1/z) for all z ∈ 𝔻 \ {0}.
D.There exists a nonzero constant c and a positive integer n such that f(z) = cz $^{n}e^(-1/z)$ for all z ∈ 𝔻 \ {0}.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Consider the entire function

f(z) = z^{2}(e

ᶻ − e⁻ᶻ) and the meromorphic function

g(z) = z^{2}/(e

ᶻ − e⁻ᶻ) on

\mathbb{C}

. Which of the following statements is true?

A.z = 0 is a zero of f of order 3 and a pole of g of order 1.
B.z = 0 is a zero of f of order 2 and a pole of g of order 1.
C.z = 0 is a zero of f of order 3 and a zero of g of order 1.✓ Correct
D.z = 0 is a zero of f of order 2 and a zero of g of order 1.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

Let

f(z) = \exp(z + 1/z), z \in \mathbb{C}

∖{0}. The residue of f at z = 0 is

A. $\sum_{l=0}^\infty$ 1/(l+1)!
B. $\sum_{l=0}^\infty$ 1/(l!(l+1))
C. $\sum_{l=0}^\infty$ 1/(l!(l+1)!)✓ Correct
D. $\sum_{l=0}^\infty$ $1/((l^{2}+l)!)$

Solution

Multiply the series eᶻ $= \sum z^{m}/m$ ! and $e^{1/z}$ $= \sum$ $z^{-k}/k$ !; the coefficient of $z^{-1}$ comes from $k = m + 1: \sum_m 1/(m!(m+1)!)$ .

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Part Bshow ▾

Let f have an essential singularity at

z_{0}

. By the Casorati–Weierstrass theorem, the image under f of every punctured neighbourhood of

z_{0}

A.dense in $\mathbb{C}$ ✓ Correct
B.all of $\mathbb{C}$
C.a bounded subset of $\mathbb{C}$
D.contained in a half-plane

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Classify the singularity at z = 0 of each function. Which statements are correct?

A.sin(z)/z has a removable singularity at 0.✓ Correct
B. $e^{1/z}$ has an essential singularity at 0.✓ Correct
C.1/(1 − cos z) has a pole of order 2 at 0.✓ Correct
D.z· $e^{1/z}$ has a pole at 0.

Solution

(1) sin z/z → 1, and a bounded isolated singularity is removable (Riemann).

(2) The Laurent series $\sum$ $z^{-n}/n$ ! has infinitely many negative-power terms.

$(3) 1 - \cos z = z^{2}/2 - z^{4}/24 + \cdots = z^{2} \times ($ analytic and nonzero at 0), so the reciprocal has a double pole.

(4) fails: z· $e^{1/z}$ = z + 1 + 1/(2!z) + ⋯ still has infinitely many negative powers — multiplying an essential singularity by zᵏ never produces a pole.

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Part Cshow ▾

Let

f(z) = 1/(z^{2}(z - 1))

. Which of the following are true?

A.z = 0 is a pole of order 2.✓ Correct
B.z = 1 is a removable singularity.
C.Res(f, 1) = 1.✓ Correct
D.Res(f, 0) = 1.

The worked solution and the per-option analysis are in Trap Analysis.

2. Residue theorem and standard contour integrals

Read this as a chapter →

Lecture - 10.2 Residue theorem

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Then drill the five standard contour patterns in the notes.

Residue theorem — the five standard patterns free

In plain words

Only the coefficient sitting on $1/z$ survives being integrated round a loop. The theorem says the integral is $2\pi i$ times the residues enclosed, turning a hard integral into arithmetic.

The theorem

$\oint_\gamma f = 2\pi{}i \sum ($ winding number) × (residues inside).

Pattern 1 — rational trigonometric: $\int_{0}^{2\pi}$ $R(\cos \theta, \sin \theta) d\theta$

Put z = $e^{i\theta}$ : $\cos \theta = (z + z$ ⁻ $^{1})/2, \sin \theta = (z - z$ ⁻ $^{1})/(2i), d\theta =$ dz/(iz). Reduces to residues in |z| < 1.

Standard result: $\int_{0}^{2\pi}$ $d\theta/(a + b \cos \theta) = 2\pi/\sqrt{a^{2} - b^{2}}$ for a > |b|.

Pattern 2 — rational over $\mathbb{R}: \int_{-\infty}^{\infty}$ P/Q dx

Close with a large semicircle in the upper half-plane; valid when deg Q ≥ deg P + 2. Answer $= 2\pi{}i \sum$ residues in the UHP.

$\int_{-\infty}^{\infty}$ dx $/(1 + x^{2}) = \pi, \int_{-\infty}^{\infty}$ dx $/(1 + x^{4}) = \pi/\sqrt{2}$ .

Pattern 3 — oscillatory: $\int f(x)$ $e^{iax}$ dx, a > 0

Same contour; Jordan's lemma kills the arc when f → 0 uniformly. Take real/imaginary parts at the end.

$\int_{0}^\infty (\sin x)/x$ dx $= \pi/2$ — needs an indentation at 0 contributing $i\pi$ ·Res $= i\pi$ .

Pattern 4 — branch cuts: $\int_{0}^\infty$ $x^{a-1}/(1$ + x) dx $= \pi/\sin(\pi{}a), 0 < a < 1$

Keyhole contour around the positive real axis.

Pattern 5 — summation of series

$\sum_{n\in\mathbb{Z}}$ $f(n) = -\sum$ residues of $\pi \cot(\pi{}z) f(z)$ at the poles of f. Gives $\sum 1/n^{2} = \pi^{2}/6$ .

Indentation rule

A small semicircular indentation of radius $\varepsilon$ around a simple pole contributes $\pm{}i\pi\cdot$ Res (sign by orientation) — half of $2\pi{}i. A$ common Part-B trap.

Key takeaways

Match the integrand to the pattern before computing anything.
Jordan's lemma is what lets you drop the big arc for $e^{iax}$ .
Indentations give $i\pi\cdot$ Res, full loops give $2\pi{}i\cdot$ Res.

Computing residues without expanding the seriesinteractive

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Simple poles, higher-order poles and the check that catches arithmetic slips.

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Exam focus: Five standard contour patterns cover almost every exam integral. Know which contour goes with which integrand, and remember that a semicircular indentation contributes iπ·Res, not 2πi·Res.

Practice

Part C2025 Juneshow ▾

Let

\gamma : [0, 1] \to \mathbb{C}

be the function t ↦

e^{2\pi{}it}

and

I = \int_\gamma e^z

e^{1/z}

dz. Which of the following statements are true?

A.I = 0
B. $(1/2\pi{}i)I \in$ { $4n : n \in \mathbb{Z}, n \ge 1$ }
C. $I = 2\pi{}i \sum_{n=0}^\infty$ 1/n!
D. $I = 2\pi{}i \sum_{n=0}^\infty$ 1/(n!(n + 1)!)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

For

t\in[0,2\pi]

, let

\gamma_{1}(t)=e^(

it) and

\gamma_{2}(t)=1+i+e^(

it). Which of the following statements are true?

A. $\int_\gamma1$ dz/(z sin z) = 0✓ Correct
B. $\int_\gamma1$ dz $/(z \sin z) = 2\pi{}i$
C. $\int_\gamma2$ dz $/(z \sin z) = 2\pi{}i$
D. $\int_\gamma2$ dz/(z sin z) = 0✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let S be a finite subset of

\mathbb{C}

containing 0. Let f :

\mathbb{C}\S

\to \mathbb{C}

be a holomorphic function which has a simple pole at 0. For R > 0, let

\gamma_R

denote the path

\gamma_R(t) = Re^(2\pi

it) for t ∈ [0,1]. Which of the following statements is necessarily true?

A.The function f(1/z) has a removable singularity at 0.
B.The function f(1/z) has an essential singularity at 0.
C.If $\int_{\gamma{}R}$ f(z)dz = 0 for some R > 0, then S is not a singleton set.✓ Correct
D.If S is not a singleton set, then $\int_{\gamma{}R}$ f(z)dz = 0 for some R > 0 such that S ∩ { $z \in \mathbb{C}$ : |z| = R} = ∅.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let

\gamma_R(t) = 2 + i + Re^(2\pi

it) for t ∈ [0,1] and R = 1, 2. Which of the following statements is true?

A. $\int_{\gamma1}$ tan(z)dz $= 2\pi{}i$
B. $\int_{\gamma1}$ tan(z)dz $= -2\pi{}i$
C. $\int_{\gamma2}$ tan(z)dz $= 2\pi{}i$
D. $\int_{\gamma2}$ tan(z)dz $= -2\pi{}i$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

For integers m, n ≥ 1, let

I_(m,n) = (1/2\pi{}i)\int_C z^{m} {\bar{z}}^{n}

dz, where C is the circle {

z \in \mathbb{C}

: |z| = 1} oriented counterclockwise. Which of the following statements is true?

A.I_(m,n) = 1 if m = n
B.I_(m,n) = 1 if m + 1 = n✓ Correct
C.I_(m,n) = 1 if m = n + 1
D.I_(m,n) = 1 if m = n + 2

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define

\gamma_r(t) =

re^(it) (where

t \in [0, 2\pi])

and

I_r = (1/2\pi{}i) \int_(\gamma_r) (z^{2} + 1) / ((z - a)(z - b))

dz. Which of the following statements is necessarily true?

A. $I_{r}$ ≠ 0 if r > max{|a|, b}
B. $I_{r}$ ≠ 0 if r < max{|a|, b}
C. $I_{r}$ = 0 if r > max{|a|, b} and |a| = b✓ Correct
D. $I_{r}$ = 0 if |a| < r < b

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

p(z) = a_{0} + a_{1}z + \cdots + a_{n}z^{n}

be a polynomial of degree ≥ 3. Suppose that p(0) = 0 and p′(0) ≠ 0. Let

f : \mathbb{C}

\ {0}

\to \mathbb{C}

be defined by

f(z) = p(z)/z^{2}

. Which of the following statements are true?

A.f has a removable singularity at z = 0.
B.f has a simple pole at z = 0.✓ Correct
C. $\int_C (p(z)/z)$ dz = 0, where C is any counterclockwise oriented smooth closed curve in $\mathbb{C}$ .✓ Correct
D. $\int_\gamma f(z)$ dz $= 2\pi$ ia $_{1}$ , where $\gamma =$ { $z \in \mathbb{C}$ : |z| = 2025} is a counterclockwise oriented circle.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

For an integer k, consider the contour integral

I_{k} = \int_{|z|=1}

eᶻ/zᵏ dz. Which of the following statements are true?

A. $I_{k} = 0$ for every integer k.
B. $I_{k} \ne 0$ if k ≥ 1.✓ Correct
C.| $I_{k}$ | ≤ | $I_{k}$ ₊ $_{1}$ | for every integer k.
D. $lim_{k\to\infty}$ | $I_{k}$ | $= \infty$ .

Solution

$I_{k} = 2\pi{}i \cdot ($ coefficient of $z^{k-1}$ in eᶻ $) = 2\pi{}i/(k-1)$ ! for k ≥ 1 and 0 for k ≤ 0.

These values decrease to 0, so (3) and (4) fail.

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Part Bshow ▾

The value of

\oint_{|z|=2}

/(z^{2} + 1)

, taken counterclockwise, is

A.0✓ Correct
B. $2\pi{}i$
C. $\pi{}i$
D. $\pi$

Solution

Both simple poles ±i lie inside |z| = 2, with residues 1/(2i) and −1/(2i) (from 1/(2z) at each pole).

They cancel, so the integral is 0 — consistent with the integrand decaying like $1/z^{2}$ at infinity.

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Part Cshow ▾

Let

f(z) = (z^{2} + 1)/(z(z - 2)^{2})

and let C be the positively oriented circle |z| = 1. Which of the following are true?

A.The only singularity of f inside C is a simple pole at 0.✓ Correct
B.Res(f, 0) = 1/4.✓ Correct
C. $\oint_C f$ dz $= \pi{}i/2$ .✓ Correct
D.Res(f, 2) = 3/4.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

The value of

\oint_{|z|=2}

/(z^{2} + 1)

A.0✓ Correct
B. $2\pi{}i$
C. $\pi{}i$
D. $-2\pi{}i$

The worked solution and the per-option analysis are in Trap Analysis.

Singularities and Residues

1. Laurent series, classification of singularities, Casorati–Weierstrass

Singularities — three types, three tests free

In plain words

The trichotomy at an isolated singularity z0z_{0}z0​

Standard examples

Computing residues

Laurent expansion depends on the annulus

Key takeaways

Classifying an isolated singularity in one lookinteractive

2. Residue theorem and standard contour integrals

Residue theorem — the five standard patterns free

In plain words

The theorem

Pattern 1 — rational trigonometric: ∫02π\int_{0}^{2\pi}∫02π​ R(cos⁡θ,sin⁡θ)dθR(\cos \theta, \sin \theta) d\thetaR(cosθ,sinθ)dθ

Pattern 2 — rational over R:∫−∞∞\mathbb{R}: \int_{-\infty}^{\infty}R:∫−∞∞​ P/Q dx

Pattern 3 — oscillatory: ∫f(x)\int f(x)∫f(x) eiaxe^{iax}eiax dx, a > 0

Pattern 4 — branch cuts: ∫0∞\int_{0}^\infty∫0∞​ xa−1/(1x^{a-1}/(1xa−1/(1 + x) dx =π/sin⁡(πa),0<a<1= \pi/\sin(\pi{}a), 0 < a < 1=π/sin(πa),0<a<1

Pattern 5 — summation of series

Indentation rule

Key takeaways

Computing residues without expanding the seriesinteractive

The trichotomy at an isolated singularity $z_{0}$

Pattern 1 — rational trigonometric: $\int_{0}^{2\pi}$ $R(\cos \theta, \sin \theta) d\theta$

Pattern 2 — rational over $\mathbb{R}: \int_{-\infty}^{\infty}$ P/Q dx

Pattern 3 — oscillatory: $\int f(x)$ $e^{iax}$ dx, a > 0

Pattern 4 — branch cuts: $\int_{0}^\infty$ $x^{a-1}/(1$ + x) dx $= \pi/\sin(\pi{}a), 0 < a < 1$