Complex Analysis, Algebra & Topology

Analytic Functions

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1. Cauchy–Riemann equations, harmonic functions

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Lecture 3.4 - Cauchy Riemann equations

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Note carefully what CR alone does and does not give you.

Lecture - 4.1 - Harmonic functions

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Harmonic conjugates and where simple connectedness is needed.

Cauchy–Riemann — what they do and do not give you free

In plain words

A real map of the plane has four derivatives to play with. A complex derivative is one number, and these two equations force four down to one — a rotation with a scaling.

The equations

f = u + iv is complex differentiable at $z_{0}$ iff f is real-differentiable there and $u_{x} = v_y, u_y = -v_{x}$ . Then $f' = u_{x} +$ iv $_{x}$ .

Hypothesis	Conclusion
CR hold at a single point + real-differentiable there	f′ exists at that point
CR hold on an open set and partials are continuous	f is holomorphic there (Goursat: continuity is automatic)
CR hold on an open set, partials merely exist	not enough

The classic spoiler: $f(z) = ({\bar{z}})^{2}/z$ for z ≠ 0, f(0) = 0 satisfies the CR equations at 0 but is not differentiable at 0 (the limit depends on the direction of approach).

Consequences worth quoting

f holomorphic with f′ ≡ 0 on a domain (connected open set) ⇒ f constant. Connectedness is essential.
Any of these forces a holomorphic f on a domain to be constant: |f| constant, Re f constant, Im f constant, arg f constant, f̄ holomorphic, f real-valued.
u and v are harmonic $(\Delta{}u = 0)$ and are harmonic conjugates; the level curves $u = c_{1}$ and $v = c_{2}$ meet orthogonally where $\nabla{}u \ne 0$ . $- \nabla{}u \cdot \nabla{}v = 0$ for holomorphic $f: u_{x}v_{x} + u_yv_y = u_{x}(-u_y) + u_yu_{x} = 0$ .
A harmonic conjugate exists on any simply connected domain. On an annulus it can fail: u = log|z| has no single-valued conjugate on $\mathbb{C}$ ∖{0}.

Anti-holomorphic traps

If f is entire then:

z ↦ conj(f(z̄)) is entire (conjugate the coefficients);
z ↦ f(z̄) and z ↦ conj(f(z)) are anti-holomorphic — entire only when f is constant.

Key takeaways

CR at a point ≠ holomorphic at a point; you need real-differentiability too.
Any "constant modulus / constant real part / conjugate also holomorphic" hypothesis collapses f to a constant.
conj(f(z̄)) is the one conjugation that preserves holomorphy.

Cauchy–Riemann: satisfied at a point, holomorphic nowhereinteractivefree

Step 1 / 4The equations

Writing $f = u+iv$ , complex differentiability at a point requires

$u_x = v_y, \qquad u_y = -v_x.$

These are necessary. They are not, on their own, sufficient.

Exam focus: CR equations alone do not give holomorphy — you need them plus continuity of the partials (or real-differentiability). The standard trap is a function satisfying CR only at the origin.

Practice

Part C2024 Decembershow ▾

For z = x + iy

\in \mathbb{C}

, let f(z) = u(x, y) + iv(x, y) be an entire function such that u(x, y) + v(x, y) = 1. Which of the following statements are FALSE?

A.f is a constant function.
B.If $f(0) \in \mathbb{R}$ , then f(0) + f(1) = 1.✓ Correct
C. $f(\mathbb{C})$ is connected.
D.If $f(2) \in \mathbb{R}$ , then f(i) + f(1) = 2.

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

Part B2024 Decembershow ▾

Let

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

be the function defined by f(z) = e^((cos(1+i)) sin z). For z = x + iy

\in \mathbb{C}

, write f(z) as u(x, y) + iv(x, y), where u, v are real-valued functions. Which of the following is the value of

(\partial{}u/\partial{}x)(0, 0)

A.0
B.(e + 1/e)(cos 1)/2✓ Correct
C.(e − 1/e)(cos 1)/2
D.1

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

Part B2024 Decembershow ▾

For z = x + iy

\in \mathbb{C}

, let f(z) = u(x, y) + iv(x, y) define an entire function. Consider the function g(z) = u(x, −y) − iv(x, −y), for z = x + iy

\in \mathbb{C}

. Suppose that v(x, 0) = 0 for all

x \in \mathbb{R}

. Define E = {

z \in \mathbb{C}

| f(z) = g(z)}. Which of the following statements is true?

A.E is the real axis.
B.E is the imaginary axis.
C.E contains an open subset of $\mathbb{C}$ , but $E \ne \mathbb{C}$ .
D. $E = \mathbb{C}$ ✓ Correct

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

Part B2024 Decembershow ▾

Let 𝔻 = {z = x + iy

\in \mathbb{C}

: |z| < 1} be the open unit disc and f : 𝔻

\to \mathbb{C} a

holomorphic function such that f(0) = 0. Let

\psi(z) =

|f(z)|

^{2}

, and

\partial^{2}\psi/\partial{}x^{2} + \partial^{2}\psi/\partial{}y^{2} \equiv 0

. Which of the following statements is FALSE?

A.f can be extended to $\mathbb{C}$ as an entire function.
B.f must have infinitely many zeros in 𝔻.
C.f is not a polynomial.✓ Correct
D.exp(f) cannot take every complex value.

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

Part C2023 Juneshow ▾

Let f(z) be an entire function on

\mathbb{C}

. Which of the following statements are true?

A.f(z̄) is an entire function.
B.conj(f(z)) is an entire function.
C.conj(f(z̄)) is an entire function.✓ Correct
D.conj(f(z̄)) + f(z̄) is an entire function.

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

Part B2023 Decembershow ▾

Let ℍ = {

z \in \mathbb{C} : Im z > 0

} and f(z) =

e^{iz}

. Which one of the following statements is true?

A.f(ℍ $) = \mathbb{C}$ ∖{0}.
B.f(ℍ) ∩ ℍ is countable.
C.f(ℍ) is bounded.✓ Correct
D.f(ℍ) is a convex subset of $\mathbb{C}$ .

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

Part B2023 Decembershow ▾

Let

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

be a real-differentiable function and define u(x, y) = Re f(x + iy), v(x, y) = Im f(x + iy). Let

\nabla{}u = (u_{x}, u_y)

denote the gradient. Which one of the following is necessarily true?

A.For $c_{1}, c_{2} \in \mathbb{C}$ , the level curves $u = c_{1}$ and $v = c_{2}$ are orthogonal wherever they intersect.
B. $\nabla{}u \cdot \nabla{}v = 0$ at every point.
C.If f is an entire function, then $\nabla{}u \cdot \nabla{}v = 0$ at every point.✓ Correct
D.If $\nabla{}u \cdot \nabla{}v = 0$ at every point, then f is an entire function.

Solution

For entire f the Cauchy–Riemann equations give $\nabla{}u \cdot \nabla{}v = u_{x}v_{x} + u_yv_y = u_{x}(-u_y) + u_yu_{x} = 0$ .

Without holomorphy there is no such relation, and orthogonal gradients alone do not force CR (e.g. f = z̄ has $\nabla{}u \cdot \nabla{}v = 0)$ .

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Part Cshow ▾

Let

u(x, y) = x^{2} - y^{2}

. Which of the following are true?

A.u is harmonic on $\mathbb{R}^{2}$ .✓ Correct
B.v(x, y) = 2xy is a harmonic conjugate of u, giving f(z) = u + iv $= z^{2}$ .✓ Correct
C.The harmonic conjugate of u is unique.
D.u attains a strict local maximum or minimum at the origin.

Solution

$(1) u_{xx} + u_yy = 2 - 2 = 0. (2)$ solving $v_{x} = -u_y = 2y, v_y = u_{x} = 2x$ gives v = 2xy (up to a constant), and u + iv $= (x^{2}-y^{2}) + i(2$ xy $) = z^{2}. (3)$ is false: any two harmonic conjugates of u differ by a real constant, so the conjugate is unique only up to an additive constant, not outright. (4) is false: non-constant harmonic functions satisfy the maximum principle and never attain a strict interior extremum — u has a saddle at the origin, not a max or min.

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Part Cshow ▾

Let f(z) = u(x, y) + iv(x, y) on a domain D. Which of the following are true?

A.If f is holomorphic on D, then $u_{x} = v_y$ and $u_y = -v_{x} ($ the Cauchy–Riemann equations).✓ Correct
B.If u, v satisfy the Cauchy–Riemann equations at a point and have continuous partial derivatives there, then f is differentiable there.✓ Correct
C.If u, v merely satisfy the Cauchy–Riemann equations at a point, with no continuity assumption on the partials, then f is necessarily differentiable there.
D.If f is holomorphic on D, then u and v are both harmonic on D.✓ Correct

Solution

(1) is the necessary condition, (2) the standard sufficient condition once continuity is added, (4) the standard consequence (real and imaginary parts of a holomorphic function are harmonic). (3) is false: the Cauchy–Riemann equations holding at a single point, with no continuity hypothesis on the partial derivatives nearby, is not enough to force differentiability there — the continuity hypothesis in (2) is doing real work.

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Part Cshow ▾

Let f(z) = u(x, y) + iv(x, y) be defined on

\mathbb{C}

. Which of the following are true?

A.If f is holomorphic on $\mathbb{C}$ , then u and v are harmonic.✓ Correct
B.If u and v satisfy the Cauchy–Riemann equations at a single point, then f is differentiable at that point.
C.f(z) = z̄ satisfies the Cauchy–Riemann equations nowhere.✓ Correct
D.If f is holomorphic and real-valued on all of $\mathbb{C}$ , then f is constant.✓ Correct

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

Part Bshow ▾

The function f(z) = |z|

^{2}

is complex differentiable

A.nowhere
B.only at z = 0✓ Correct
C.everywhere
D.exactly on the real axis

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

2. Power series and analyticity

Read this as a chapter →

Lecture 3.1- Power series

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Lecture 3.2 - Differentiation of Power series

NPTEL · Complex Analysis (Pranav Haridas, KSoM)

Holomorphic ⇔ analytic — the equivalence with no real analogue.

Analyticity, the identity theorem and zeros free

In plain words

In real analysis, smooth and equal-to-its-Taylor-series are different. In complex analysis they are the same thing, and the radius is simply the distance to the nearest point where the function breaks.

The great equivalence

For f on an open set $\Omega$ the following are equivalent: (i) f is complex differentiable on $\Omega; ($ ii) f is infinitely differentiable; (iii) around every point f equals a convergent power series; (iv) f is continuous and $\oint_\gamma f = 0$ over every triangle (Morera).

There is no real analogue: $e^{-1/x^{2}}$ is $C^\infty$ on $\mathbb{R}$ but not real-analytic at 0.

Cauchy estimates and their children

|f⁽ $^{n}$ ⁾ $(z_{0})$ | $\le n! M / r^{n}$ where M = sup on | $z - z_{0}$ | = r. From this:

Liouville: bounded entire ⇒ constant. Variants: Re f bounded ⇒ constant (apply Liouville to e^f); |f| ≤ C|z|ᵏ ⇒ polynomial of degree $\le k; f(z) \to \infty$ as |z| $\to \infty \Rightarrow$ polynomial.
Fundamental theorem of algebra: else 1/p is bounded entire.

Zeros and the identity theorem

Zeros of a non-constant holomorphic function on a domain are isolated and of finite order. Hence:

If f = g on a set with a limit point in the domain, then f ≡ g.
sin(1/z) has zeros $1/(n\pi)$ accumulating at 0 — but 0 is not in the domain $\mathbb{C}$ ∖{0}, so no contradiction. This is the recurring Part-C trap.
A non-zero f can still have infinitely many zeros (sin z), just not accumulating inside.

Hurwitz

If $f_{n} \to f$ locally uniformly, f ≢ 0, and each $f_{n}$ is zero-free on a domain, then f is zero-free. Used for: the partial sums $\sum_{k\le{}n}$ zᵏ/k! → eᶻ have zeros escaping to infinity.

Standard radii

Series	R
$\sum z^{n}/n$ !	$\infty$
$\sum n! z^{n}$	0
$\sum z^{n}/n^p$	1
$\sum$ $z^{n!}$	1 (natural boundary on \|z\| = 1)

Key takeaways

Holomorphic = analytic; $C^\infty$ is strictly weaker over $\mathbb{R}$ .
Identity theorem needs the limit point inside the domain — check that first.
Liouville has many disguises: bounded real part, polynomial growth, no zeros of eᶠ.

Why the radius of convergence is set by the nearest singularityinteractive

This visual is part of Trap Analysis

A function smooth on all of ℝ whose series stops at |x| = 1 — explained by a pole you cannot see on the real line.

Unlock interactive visuals

Exam focus: Holomorphic ⇔ analytic ⇔ locally a convergent power series — an equivalence with no real-analysis analogue. Use the identity theorem to force f ≡ g from agreement on a set with a limit point inside the domain.

Practice

Part C2025 Juneshow ▾

Let f be an entire function which is not a polynomial. Let A = {

\alpha \in \mathbb{C}

| f⁽

^{n}

⁾

(\alpha) \ne 0

for all n ≥ 0}. Which of the following statements are true?

A.A is nonempty.✓ Correct
B.A is finite.
C.A is infinite.✓ Correct
D.A is uncountable.✓ Correct

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

Part C2025 Decembershow ▾

Consider the following statements: (P) The initial value problem

y'+y=e^(-x^{2}), y(0)=0

has a Taylor series solution about the point x=0. (Q) The initial value problem y′+y=r(x), y(0)=0, where

r(x)=e^(-1/x^{2})

for x≠0 and r(0)=0, has a Taylor series solution about the point x=0. Then which of the following statements are true?

A.(P) is true.✓ Correct
B.(P) is FALSE.
C.(Q) is true.
D.(Q) is FALSE.✓ Correct

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

Part B2025 Decembershow ▾

Let 𝔻 denote the open unit disc {

z \in \mathbb{C}

: |z| < 1} and X = {f : 𝔻

\to \mathbb{C}

∣ f is holomorphic and satisfies

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

for all |z| < 1/2}. Which of the following statements is true?

A.X is uncountable.
B.X is infinite and countable.
C.Every element of X is an open map.
D.X has exactly one element.✓ Correct

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

Part B2025 Juneshow ▾

Which of the following statements is true?

A.There exists an entire function f such that f⁽ $^{n}$ ⁾ $(0) = n!/n^{n}$ for all positive integers n.✓ Correct
B.There exists an entire function f such that f⁽ $^{n}$ ⁾ $(0) = n!n^{n}$ for all positive integers n.
C.There exists an entire function f such that f⁽ $^{n}$ ⁾(0) = (n − 1)! for all positive integers n.
D.There exists an entire function f such that f⁽ $^{n}$ ⁾(0) = n!n for all positive integers n.

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

Part C2024 Juneshow ▾

Which of the following conditions ensure that the power series

\sum_(n\ge0) a_{n}z^{n}

defines an entire function?

A.The power series converges for every $z \in \mathbb{C}$ ✓ Correct
B.The power series converges for every $z \in \mathbb{R}$ ✓ Correct
C.The power series converges for every z ∈ { $2^{n} : n \in \mathbb{N}$ }✓ Correct
D.The power series converges for every z ∈ { $1/5^{n} : n \in \mathbb{N}$ }

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

Part C2024 Decembershow ▾

Let

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

be an entire function such that f(z) = f(iz) for all

z \in \mathbb{C}

. Which of the following statements are true?

A.f(z) = f(−z) for all $z \in \mathbb{C}$ .✓ Correct
B.f′(0) = f″(0) = f‴(0) = 0✓ Correct
C.There is an entire function $g : \mathbb{C} \to \mathbb{C}$ such that $f(z) = g(z^{4})$ for all $z \in \mathbb{C}$ .✓ Correct
D.f is necessarily a constant function.

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

Part C2023 Decembershow ▾

For every n ≥ 1, consider the entire function

p_{n}(z) = \sum_{k=0}^{n}

zᵏ/k!. Which of the following statements are true?

A.The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $\mathbb{C}$ .✓ Correct
B.For all $n \ge 1, p_{n}$ has a zero in the set { $z \in \mathbb{C}$ : |z| ≤ 2023}.
C.There exists a sequence $(z_{n})$ of complex numbers such that lim | $z_{n}$ | $= \infty$ and $p_{n}(z_{n}) = 0$ for all n ≥ 1.✓ Correct
D.Let $S_{n}$ be the set of all zeros of $p_{n}$ . If $a_{n} =$ $min_{z\in{}S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$ .✓ Correct

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

Part B2023 Juneshow ▾

Consider the function f defined by

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

for

z \in \mathbb{C}

such that

1 - z - z^{2} \ne 0

. Which of the following statements is true?

A.f is an entire function.
B.f has a simple pole at z = 0.
C.f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$ , where $a_{0} = 1, a_{1} = 0$ and $a_{n}$ ₊ $_{2} = a_{n} + a_{n}$ ₊ $_{1}$ for n ≥ 0.
D.f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$ , where $a_{0} = 1, a_{1} = 1$ and $a_{n}$ ₊ $_{2} = a_{n} + a_{n}$ ₊ $_{1}$ for n ≥ 0.✓ Correct

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