Complex Analysis, Algebra & Topology

Cauchy Theory

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1. Cauchy's theorem and integral formula

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Lecture - 6.2 Cauchy-Goursat theorem

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Lecture - 6.3 Cauchy's theorem

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Watch where simple connectedness enters — that is the whole point.

Lecture - 7.1 Cauchy Integral Formula

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Cauchy's theorem and integral formula — the toolkit free

In plain words

Round a closed loop a well-behaved function integrates to zero, provided the region inside has no holes. The single exception, $1/z$ round the origin, is where all the content lives.

Statements

Cauchy–Goursat: f holomorphic on a simply connected $\Omega \Rightarrow \oint_\gamma f = 0$ for every closed $\gamma$ in $\Omega$ .
Integral formula: $f(z_{0}) = (1/2\pi{}i) \oint_\gamma f(z)/(z - z_{0})$ dz for $\gamma$ winding once around $z_{0}$ .
Derivatives: f⁽ $^{n}$ ⁾ $(z_{0}) = (n!/2\pi{}i) \oint_\gamma f(z)/(z - z_{0})^{n+1}$ dz.

Why simple connectedness matters

$\oint_{|z|=1}$ dz $/z = 2\pi{}i \ne 0$ although 1/z is holomorphic on $\mathbb{C}$ ∖{0} — the domain is not simply connected. This single integral is the seed of all residue theory.

Choosing the tool

Integrand inside $\gamma$	Method
no singularity	Cauchy's theorem: 0
one simple pole	integral formula
a pole of order n	derivative formula
several poles	residue theorem
essential singularity	Laurent coefficient a₋ $_{1}$

Worked patterns

$- \oint_{|z|=2}$ dz $/(z^{2} + 1) = 0$ — residues at ±i are 1/(2i) and −1/(2i), which cancel. (Zero without Cauchy's theorem applying!) $- \oint_{|z|=1}$ eᶻ/zᵏ dz $= 2\pi{}i/(k-1)$ ! for k ≥ 1, and 0 for k ≤ 0. $- \oint_{|z|=r}$ $z^{n}$ dz $= 2\pi{}i$ if n = −1, else 0 — the only integral you really need.

Consequences

Mean value property: $f(z_{0})$ is the average of f over any circle centred at $z_{0}$ .
Maximum modulus: a non-constant holomorphic function on a domain has no interior maximum of |f|; on a bounded domain the max of |f| is attained on the boundary.
Minimum modulus: same for |f| minimum provided f has no zeros.
Schwarz lemma: f : 𝔻 → 𝔻 holomorphic with f(0) = 0 ⇒ |f(z)| ≤ |z| and |f′(0)| ≤ 1, with equality only for a rotation.

Key takeaways

$- \oint$ dz $/z = 2\pi{}i$ is the whole story; everything else is bookkeeping.

Vanishing of an integral does not mean the theorem applies — residues can just cancel.
Maximum modulus pushes extrema to the boundary; that is how "bounded on the boundary" arguments work.

Cauchy's integral formula, and what it gives you for freeinteractivefree

Step 1 / 4The theorem

If $f$ is holomorphic on and inside a simple closed contour $C$ , then $\oint_C f = 0$ . If $a$ is inside $C$ ,

$f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz.$

The value at an interior point is determined entirely by the boundary values.

Exam focus: Pick the right tool: Cauchy's theorem (no singularities inside) vs the integral formula (one simple pole) vs residues (several). Simple connectedness is what makes ∮ = 0.

Practice

Part B2024 Decembershow ▾

Let ℍ = {z = x + iy

\in \mathbb{C}

| y > 0} and f : ℍ

\to \mathbb{C}

be a non-constant holomorphic function satisfying |f(z)| < 1 for all z ∈ ℍ. Which of the following statements is true?

A. $\lim(y\to+\infty) f'($ iy) = 0✓ Correct
B. $\lim(y\to+\infty) f'($ iy) is a complex number with absolute value 1.
C. $\lim(y\to+\infty)$ |f′(iy)| $= +\infty$
D. $\lim(y\to+\infty) f'($ iy) is not a real number.

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

Part B2024 Juneshow ▾

Consider the contour

\gamma

given by

\gamma(\theta) = e^(2i\theta)

for

\theta \in [0, \pi/2]; \gamma(\theta) = 1 + 2e^(2i\theta)

for

\theta \in [\pi/2, 3\pi/2]; \gamma(\theta) = e^(2i\theta)

for

\theta \in [3\pi/2, 2\pi]

. Then what is the value of

\int_\gamma

dz / (z(z − 2))?

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

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

Part B2024 Decembershow ▾

Let

\gamma

be the circle {

z \in \mathbb{C}

: |z| = 3} oriented counterclockwise. Let f be an entire function. What is the value of A for which

\int_\gamma (A/(z - 1) - f(z)/(z - 2)^{2})

dz = 0 holds?

A.f(1)
B.f(2)
C.f′(1)
D.f′(2)✓ Correct

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

Part Cshow ▾

Let f be entire and let

\gamma_R

be the circle |z| = R. Which of the following are true?

A.f⁽ $^{n}$ ⁾ $(0) = (n!/2\pi{}i) \oint_{\gamma_R}$ f(z)/z^(n+1) dz for every R > 0.✓ Correct
B.Cauchy's estimate gives |f⁽ $^{n}$ ⁾(0)| $\le n!\cdot{}M_R/R^{n}$ , where M_R = max on |z| = R of |f(z)|.✓ Correct
C.If f is entire and bounded, then f is constant (Liouville's theorem).✓ Correct
D. $\oint_{\gamma_R}$ f(z) dz depends on R in general, even for f entire.

Solution

(1) is Cauchy's formula for derivatives, (2) the ML-inequality consequence known as Cauchy's estimate, (3) Liouville's theorem. (4) is false: f entire has no singularities anywhere, so Cauchy's theorem gives $\oint_{\gamma_R}$ f(z) dz = 0 for every R — the value does not depend on R at all, since there is never an obstruction inside any circle.

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Part Cshow ▾

Let

f(z) = 1/(z^{2} + 4)

and let

\gamma

be the circle |z| = 3, traversed once counterclockwise. Which of the following are true?

A.Both singularities of f lie inside $\gamma$ .✓ Correct
B. $\oint_\gamma f(z)$ dz = 0.✓ Correct
C.f is holomorphic everywhere inside $\gamma$ .
D.The value of the integral would change if $\gamma$ were replaced by |z| = 10.

Solution

f has simple poles at ±2i, both inside |z| = 3, confirming (1).

The residues there are 1/(4i) and −1/(4i), which sum to 0, so by the residue theorem the integral is 0 — (2) holds.

(3) is false: f is not holomorphic at ±2i, which sit inside $\gamma$ .

(4) is false: |z| = 10 still encloses exactly the same two poles and nothing else, and the deformation theorem says the integral depends only on which singularities are enclosed, not on the contour's exact shape — it stays 0.

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Part Cshow ▾

Let f be holomorphic on a domain

D \subseteq \mathbb{C}

and let

\gamma

be a simple closed contour in D whose interior lies in D. Which of the following are true?

A. $\oint_\gamma f(z)$ dz = 0, by Cauchy's theorem.✓ Correct
B.For every a inside $\gamma, f(a) = (1/2\pi{}i) \oint_\gamma f(z)/(z - a)$ dz.✓ Correct
C.f is infinitely differentiable on D.✓ Correct
D. $\oint_\gamma f(z)$ dz = 0 even if f has a simple pole inside $\gamma$ .

Solution

(1) is Cauchy's theorem, (2) the Cauchy integral formula, (3) the standard consequence that holomorphic implies analytic implies $C^\infty$ .

(4) is false: both results need f holomorphic throughout the interior of $\gamma$ .

A simple pole inside breaks that hypothesis entirely, and the integral instead picks up $2\pi{}i$ times the residue there, which is generally nonzero.

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Part Cshow ▾

Let f(z) = sin(z)/z for z ≠ 0, with f(0) = 1. Which of the following are true?

A.f is entire.✓ Correct
B. $\oint_{|z|=1}$ f(z) dz = 0.✓ Correct
C.f has a simple pole at z = 0.
D.f(z) = 0 has infinitely many solutions in $\mathbb{C}$ , namely $z = n\pi$ for nonzero integers n.✓ Correct

Solution

$\sin(z)/z = 1 - z^{2}/6 + z^{4}/120 - \cdots$ has no negative powers of z, so the singularity at 0 is removable and, once filled in with f(0) = 1, f is entire — (1). (2) follows from Cauchy's theorem applied to an entire function. (4) sin z = 0 exactly at $z = n\pi$ , and z = 0 gives f(0) = 1 ≠ 0, so the zeros of f are $z = n\pi$ for n ≠ 0. (3) is false — it directly contradicts (1): a genuine pole could never be patched by assigning one finite value, which is exactly why this singularity is called removable rather than a pole.

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Part Cshow ▾

Let f be entire. Which of the following are true?

A.If f is bounded then f is constant✓ Correct
B.If Re f is bounded then f is constant✓ Correct
C.If $f(z) \to \infty$ as |z| $\to \infty$ then f is a polynomial✓ Correct
D.If f has no zeros then f is constant

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

2. Liouville, Morera, maximum modulus principle

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Lecture - 7.2 Principle of analytic continuation and Cauchy estimates

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Cauchy estimates are the engine behind Liouville.

Lecture - 7.3 Further consequences of Cauchy Integral Formula

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Liouville and the fundamental theorem of algebra.

Lecture 8.2 - Open mapping theorem

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Open mapping ⇒ maximum modulus principle.

Liouville, Morera and the maximum principle free

In plain words

A complex function well behaved on the whole plane and never large has to be constant. Watch what bounded means: $\sin z$ stays small on the real line and explodes off it.

Liouville and its family

Hypothesis on entire f	Conclusion
\|f\| bounded	constant
Re f bounded above	constant (consider e^f)
Im f bounded	constant
\|f(z)\| ≤ C(1 + \|z\|ᵏ)	polynomial of degree ≤ k
$f(z) \to \infty$ as \|z\| $\to \infty$	polynomial
f omits two values	constant (Picard's little theorem)
f omits one value	not constant: eᶻ omits 0
f injective entire	f(z) = az + b, a ≠ 0

Liouville needs all of $\mathbb{C}. e$ ᶻ is bounded on any left half-plane without being constant.

Morera

f continuous on $\Omega$ with $\oint_T f = 0$ for every triangle $T \subset \Omega \Rightarrow f$ holomorphic. This is the standard route to: locally uniform limits of holomorphic functions are holomorphic, and $f(z) = \int g(z, t)$ dt is holomorphic.

Maximum modulus principle

Non-constant holomorphic f on a domain $\Omega$ :

|f| has no local maximum in $\Omega$ ;
if $\Omega$ is bounded and f extends continuously to ${\bar{\Omega}}$ , then $max_{\bar{\Omega}}$ |f| is attained on $\partial\Omega$ ;
the same for min |f| if f is zero-free (else f(z) = z on the disc is a counterexample: minimum 0 at the centre).

Schwarz–Pick / Schwarz lemma is the quantitative version on the unit disc, and gives that the automorphisms of 𝔻 are exactly the Möbius maps $e^{i\theta}(z$ − a)/(1 − āz).

Open mapping theorem

A non-constant holomorphic map on a domain is open. Consequences: |f| cannot have an interior max; a holomorphic map into $\mathbb{R}$ is constant; an injective holomorphic map has f′ ≠ 0 everywhere and a holomorphic inverse.

Key takeaways

Growth control on all of $\mathbb{C} \Rightarrow$ polynomial; boundedness ⇒ constant.
Morera is the converse-of-Cauchy tool for proving holomorphy of limits and integrals.
Minimum modulus needs zero-freeness; maximum modulus never does.

Liouville, and the growth bounds that do not force constancyinteractive

This visual is part of Trap Analysis

Four hypotheses — which collapse the function to a constant and which only make it a polynomial.

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Exam focus: Recognise Liouville in disguise: bounded, bounded real part, bounded on a growth scale, or missing two values. Maximum modulus turns interior bounds into boundary bounds.

Practice

Part B2025 Juneshow ▾

Let

f : \mathbb{C} \to \mathbb{C}

be a polynomial map. For R > 0, let

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

be the map t ↦

Re^{2\pi{}it}

. Suppose that there exists

c \in \mathbb{R}

such that

\int_{0}^{1}

|(f ∘

\gamma_R)(t) \gamma'_R(t)

| dt → c as

R \to \infty

. Which of the following statements is FALSE?

A.The function zf(1/z) → 0 as |z| $\to \infty$ .
B.The function f is constant.
C.c = 0.
D.c > 0.✓ Correct

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

Part C2025 Juneshow ▾

Let f be an entire function. Which of the following statements are true?

A.If f(z) = f(z + 1) for all $z \in \mathbb{C}$ then f is a constant function.
B.If f(z) = f(z + 1) = f(z + i) for all $z \in \mathbb{C}$ then f is a constant function.✓ Correct
C.If f(1/z) has a removable singularity at 0 then f is a constant function.✓ Correct
D.If f is a non-constant function then f(1/z) has a pole at 0.

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

Part C2025 Decembershow ▾

Let

f:\mathbb{C}\to\mathbb{C}

be defined by

f(z)=\sin^{2}z+\cos^{2}

|z|. Which of the following statements are true?

A.f is a real valued function.
B.f(z) = 1 for all $z\in\mathbb{C}$ .
C.f is not an entire function.✓ Correct
D.f has finitely many zeros on the imaginary axis.✓ Correct

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

Part B2025 Juneshow ▾

Let f be an entire function such that

f(\mathbb{C}) \subset

{x + iy | y = x + 1}. Which of the following statements is true?

A.|f(z)| $\to \infty$ as |z| $\to \infty$ .
B.f(z)/z → 0 as |z| $\to \infty$ .✓ Correct
C.zf(z) → 0 as |z| $\to \infty$ .
D.f(z) → 0 as |z| $\to \infty$ .

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

Part C2025 Decembershow ▾

Let f be an entire function. Consider the function g given by g(z) = f(z) − 1/z for

z\in\mathbb{C}

{0}. Which of the following statements are necessarily true?

A.The function g has a pole at 0.✓ Correct
B.If $g(\alpha)=0$ , then | $\alpha$ |≠1.
C.The function g has only finitely many zeros.
D. $max_{|z|=1}$ |g(z)| ≥ 1✓ Correct

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

Part C2024 Juneshow ▾

Suppose that f is an entire function such that |f(z)| ≥ 2024 for all

z \in \mathbb{C}

. Which of the following statements are necessarily true?

A.f(z) = 2024 for all $z \in \mathbb{C}$
B.f is a constant function✓ Correct
C.f is an injective function
D.f is a bijective function

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

Part B2024 Juneshow ▾

Let f be an entire function. Which of the following statements is FALSE?

A.If Re(f), Im(f) are bounded then f is constant
B.If e^(|Re(f)| + |Im(f)|) is bounded, then f is constant
C.If the sum Re(f) + Im(f) and the product Re(f)Im(f) are bounded, then f is constant
D.If sin(Re(f) + Im(f)) is bounded, then f is constant✓ Correct

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

Part C2024 Juneshow ▾

Let f be an entire function such that for every integer k ≥ 1 there is an infinite set X_k such that f(z) = 1/k for all z ∈ X_k. Which of the following statements are necessarily true?

A.There exists an infinite set X such that f(z) = 0 for all z ∈ X
B.There exists a non-empty closed set X such that f(z) = 0 for all z ∈ X
C.The set X_k is unbounded for each k ≥ 1✓ Correct
D.If there exists a bounded sequence (z_k)_(k≥1) such that z_k ∈ X_k for each k ≥ 1, then f has a zero✓ Correct

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

Part C2024 Decembershow ▾

Let

f : \mathbb{C}

\ {0}

\to \mathbb{C}

be a non-zero holomorphic function such that |f(z)| ≤ |z|^(5/2) + 1/|z|

^(1/2), z \in \mathbb{C}

\ {0}. Which of the following statements are true?

A.f has a pole at z = 0.
B.There is an entire function g such that f = g on $\mathbb{C}$ \ {0}.✓ Correct
C.f is a polynomial of degree at most 2.✓ Correct
D.f has an essential singularity at z = 0.

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

Part C2024 Decembershow ▾

Let

f : \mathbb{C}

\ {−1, 1}

\to \mathbb{C}

be a holomorphic function that does not take any value in the set {

z \in \mathbb{C}

: |z − 1| < 1}. Which of the following statements are true?

A.f is constant.✓ Correct
B.f has removable singularities at −1 and 1.✓ Correct
C.f is bounded.✓ Correct
D.f has either poles or essential singularities at −1 and 1.

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

Part C2023 Decembershow ▾

Let

\Omega_{1} =

{

z \in \mathbb{C}

: |z| < 1} and

\Omega_{2} = \mathbb{C}

. Which of the following statements are true?

A.There exists a holomorphic surjective map $f : \Omega_{1} \to \Omega_{2}$ .✓ Correct
B.There exists a holomorphic surjective map $f : \Omega_{2} \to \Omega_{1}$ .
C.There exists a holomorphic injective map $f : \Omega_{1} \to \Omega_{2}$ .✓ Correct
D.There exists a holomorphic injective map $f : \Omega_{2} \to \Omega_{1}$ .

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

Part C2023 Decembershow ▾

Let X be an uncountable subset of

\mathbb{C}

and let

f : \mathbb{C} \to \mathbb{C}

be an entire function. Assume that for every z ∈ X there exists an integer n ≥ 1 such that

f^{(n)}(z)

= 0. Which of the following statements are necessarily true?

A.f = 0.
B.f is a constant function.
C.There exists a compact subset K of $\mathbb{C}$ such that $f^{-1}(K)$ is not compact.
D.f is a polynomial.✓ Correct

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

Part Bshow ▾

Let f be entire with |f(z)| ≥ 1 for all

z \in \mathbb{C}

. Then

A.f must be constant.✓ Correct
B.f must be a polynomial of positive degree.
C.f must have a zero.
D.no such f exists.

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

Part Cshow ▾

Let f be entire. Which of the following conditions, each on its own, force f to be constant?

A.|f(z)| ≤ 5 for all $z \in \mathbb{C}$ .✓ Correct
B.Re f(z) > 0 for all $z \in \mathbb{C}$ .✓ Correct
C.|f(z)| ≤ $|z|^{3/2}$ for all |z| ≥ 1.
D.f(1/n) = 0 for every positive integer n.✓ Correct

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

Part Cshow ▾

Let f be holomorphic on the open unit disc D and continuous on its closure, with |f(z)| = 3 for every |z| = 1. Which of the following are possible?

A.f is constant with |f| = 3.✓ Correct
B.f is non-constant and has no zeros in D.
C.f is non-constant and has at least one zero in D.✓ Correct
D.|f(0)| > 3.

Solution

Maximum modulus gives |f| ≤ 3 on D, so (4) is impossible.

If f never vanishes, the minimum modulus principle (maximum modulus applied to 1/f) gives |f| ≥ 3 inside too, forcing |f| ≡ 3 and f constant — so (2) is impossible, and every non-constant example must vanish somewhere: f(z) = 3z realises (3).

(1) is realised by f ≡ 3.

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