The function f(z) = |z|² is

complex differentiable only at z = 0 and analytic nowhere

Which statements about entire functions are true?

a bounded entire function is constant

an entire function f with Re f(z) ≤ 5 for all z is constant

an entire function f with |f(z)| ≤ 1 + |z|² for all z is a polynomial of degree at most 2

a bounded entire function is constant

an entire function f with Re f(z) ≤ 5 for all z is constant

an entire function f with |f(z)| ≤ 1 + |z|² for all z is a polynomial of degree at most 2

Which statements are true?

there is an analytic f on the unit disc with f(1/n) = 1/n² for every n ≥ 2

an entire function that vanishes at every real number is identically zero

there is an analytic f on the unit disc with f(1/n) = 1/n² for every n ≥ 2

there is an analytic f on the unit disc with f(1/n) = (−1)ⁿ/n for every n ≥ 2

an entire function that vanishes at every real number is identically zero

an analytic function on the unit disc with infinitely many zeros there is identically zero

Which statements are true? (All circles are positively oriented.)

if ∮_{C} f dz = 0 for one closed curve C, then f is analytic inside C

Analytic Functions, Cauchy’s Theorem and Formula, and Power Series

Complex Analysis is Section 4 of the GATE Mathematics (MA) paper, and it rewards a small number of theorems applied with precision. This first chapter builds the theory of analytic functions: complex differentiability and the Cauchy–Riemann equations, harmonic functions and their conjugates, Cauchy’s integral theorem and formula with the formula for derivatives, and the global consequences that make analytic functions so rigid — Liouville’s theorem, the maximum modulus principle, Morera’s theorem, isolated zeros and the identity theorem — ending with power series, Taylor series and Laurent series. The single most useful fact for the paper is that an analytic function equals its Taylor series on the largest disc that fits inside its domain, so the radius of convergence is the distance to the nearest singularity, and the most useful habit is to ask, for every contour integral, which singularities lie inside the contour. The second chapter takes singularities, residues, Rouché’s theorem, the argument principle, Schwarz’s lemma and conformal maps.

1. Complex differentiability and the Cauchy–Riemann equations

f = u + iv is complex differentiable at z₀ if lim_h→0 [f(z₀ + h) − f(z₀)]/h exists, h complex. Then uₓ = v_y and u_y = −vₓ at z₀ (the Cauchy–Riemann equations) and f′ = uₓ + ivₓ. Conversely, if u and v have continuous first partials near z₀ satisfying CR at z₀, f is differentiable at z₀. f is analytic (holomorphic) at z₀ if it is differentiable on a neighbourhood of z₀; entire if analytic on all of ℂ.

Differentiable at a point is not analytic
f(z)	Cauchy–Riemann	Verdict
z̄ = x − iy	uₓ = 1 ≠ −1 = v_y	differentiable nowhere
\|z\|² = x² + y²	uₓ = 2x = v_y = 0 and u_y = 2y = −vₓ = 0 only at 0	differentiable only at 0, analytic nowhere
eᶻ, sin z, cos z, polynomials	hold everywhere	entire
Re z, Im z, \|z\|	fail on an open set	analytic nowhere; a real-valued analytic function on a domain is constant

2. Harmonic functions and harmonic conjugates

If f = u + iv is analytic then u and v are C^∞ and harmonic: uₓₓ + u_yy = 0. v is a harmonic conjugate of u. On a simply connected domain every harmonic u has a conjugate, unique up to an additive constant, found by integrating the CR equations. The conjugate of v is −u (since −if = v − iu is analytic).

Worked example. u = x³ − 3xy² + 2y. uₓ = 3x² − 3y² = v_y gives v = 3x²y − y³ + φ(x); then u_y = −6xy + 2 = −vₓ = −6xy − φ′(x), so φ′ = −2 and v = 3x²y − y³ − 2x + c. Indeed f = z³ − 2iz.

⚠️ Simple connectivity matters for conjugates

u = log|z| = ½ log(x² + y²) is harmonic on ℂ ∖ {0}, but its conjugate would be arg z, which cannot be defined continuously round the origin. So a harmonic function on a non-simply-connected domain need not be the real part of an analytic function there; locally it always is.

3. Cauchy’s integral theorem and formula, and Morera’s theorem

Cauchy’s theorem. If f is analytic on a simply connected domain D, then ∮_γ f dz = 0 for every closed piecewise smooth curve γ in D; equivalently f has a primitive on D. Cauchy’s integral formula. If f is analytic on and inside a positively oriented simple closed curve γ and a is inside, then f(a) = (1/2πi)∮_γ f(z)/(z − a) dz, and more generally f⁽ⁿ⁾(a) = (n!/2πi)∮_γ f(z)/(z − a)ⁿ⁺¹ dz. So analytic functions are infinitely differentiable.

∮_|z|=2 eᶻ/(z − 1)³ dz = (2πi/2!)·(eᶻ)″|_z=1 = πie.
∮_|z−i|=1 dz/(z² + 1): only z = i is inside, so write 1/(z² + 1) = [1/(z + i)]/(z − i) and get 2πi · 1/(2i) = π.
∮_|z|=1 z̄ dz is not covered by Cauchy (z̄ is not analytic): on the circle z̄ = 1/z, so the integral is 2πi.

Morera’s theorem is the converse: if f is continuous on a domain D and ∮_∂T f dz = 0 for every triangle T in D, then f is analytic on D. It is how one proves that uniform limits of analytic functions are analytic.

4. Liouville, maximum modulus, and zeros

Cauchy’s estimate: if |f| ≤ M on |z − a| = R then |f⁽ⁿ⁾(a)| ≤ n!M/Rⁿ. Liouville: a bounded entire function is constant. So an entire f with |f(z)| ≤ C(1 + |z|ⁿ) is a polynomial of degree ≤ n, and an entire f with Re f ≤ M is constant (apply Liouville to e^f). Liouville proves the fundamental theorem of algebra.
Maximum modulus: a non-constant analytic f on a domain has no local maximum of |f|; on a bounded domain with f continuous up to the boundary, max |f| is attained on the boundary. If f has no zeros, the same holds for the minimum. For f = z² + 2z on |z| ≤ 1, |f| = |z||z + 2| ≤ 3, attained at z = 1.
Zeros are isolated for a non-constant analytic function on a domain, and each has a finite order m: f = (z − a)ᵐg with g(a) ≠ 0. Identity theorem: if f = g on a set with a limit point in the domain, then f ≡ g there.

⚠️ The limit point must lie inside the domain

sin(π/(1 − z)) is analytic on the unit disc and vanishes at every 1 − 1/k, infinitely many points — but they accumulate at 1, on the boundary, so the identity theorem says nothing and the function is not zero. By contrast, f(1/n) = 1/n² forces f(z) = z² on the disc, and no analytic f can satisfy f(1/n) = (−1)ⁿ/n, since even n would force f = z and odd n would force f = −z.

5. Power series, Taylor series and Laurent series

A power series Σaₙ(z − a)ⁿ converges absolutely in the open disc |z − a| < R, R = 1/limsup |aₙ|^1/n, uniformly on smaller closed discs, and its sum is analytic there. Conversely Taylor’s theorem: if f is analytic on |z − a| < R, then f(z) = Σ f⁽ⁿ⁾(a)(z − a)ⁿ/n! throughout that disc. So the radius of convergence of the Taylor series about a is the distance from a to the nearest point where f fails to be analytic.

Radius from the nearest singularity
f, expanded about 0	Nearest singularity	R
1/(1 + z²)	±i (although smooth on all of ℝ)	1
1/(z² + 2z + 2)	−1 ± i	√2
tan z	±π/2	π/2
z/(eᶻ − 1)	±2πi (z = 0 is removable)	2π

Laurent’s theorem. If f is analytic on the annulus r < |z − a| < R, then f(z) = Σ_n=−∞^∞ cₙ(z − a)ⁿ there, with cₙ = (1/2πi)∮ f(ζ)/(ζ − a)ⁿ⁺¹ dζ over any circle in the annulus; the expansion depends on the annulus. For f = 1/((z − 1)(z − 2)) = 1/(z − 2) − 1/(z − 1) on 1 < |z| < 2: 1/(z − 2) = −½Σ(z/2)ⁿ and −1/(z − 1) = −Σ z⁻ⁿ⁻¹, so c₂ = −1/8 and c₋₁ = −1.

Key takeaways

Cauchy–Riemann with continuous partials gives differentiability; analytic means differentiable on an open set, so |z|² is differentiable at 0 and analytic nowhere.
Real and imaginary parts of analytic functions are harmonic; conjugates come from integrating CR and exist globally on simply connected domains.
Cauchy’s formula f⁽ⁿ⁾(a) = (n!/2πi)∮f/(z − a)ⁿ⁺¹ turns contour integrals into derivatives at the enclosed point.
Liouville, maximum modulus and the identity theorem make analytic functions rigid; the identity theorem needs a limit point inside the domain.
The Taylor radius is the distance to the nearest singularity; Laurent coefficients depend on the annulus.

Practice questions (13)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

The function f(z) = |z|² is
1. complex differentiable only at z = 0 and analytic nowhere
2. entire
3. analytic only at z = 0
4. differentiable nowhere
Show answer
Answer: A — complex differentiable only at z = 0 and analytic nowhere
u = x² + y², v = 0: the CR equations 2x = 0 and 2y = 0 hold only at the origin, and the partials are continuous, so f is differentiable there only. Analytic at a point requires differentiability on a neighbourhood, which fails at every point including 0.
Which of the following functions is entire?
1. cos z
2. z̄
3. |z|
4. Re z
Show answer
Answer: A — cos z
cos z = (e^iz + e^−iz)/2 is analytic on ℂ. z̄ fails CR everywhere; |z| and Re z are real-valued and non-constant, and a real-valued analytic function on a domain must be constant (CR forces all partials of u to vanish when v = 0).
Let u(x, y) = x³ − 3xy² + 2y and let v be the harmonic conjugate of u with v(0, 0) = 0. The value of v(1, 2) is ____.
Numerical answer — type the value.
Show answer
Answer: -4
From uₓ = v_y: v = 3x²y − y³ + φ(x). From u_y = −vₓ: −6xy + 2 = −6xy − φ′, so φ = −2x + c, and v(0, 0) = 0 gives c = 0. v(1, 2) = 3·1·2 − 8 − 2 = −4. Check: f = z³ − 2iz gives f(1 + 2i) = −7 − 4i. Typed, the answer is -4.
Which of the following functions are harmonic on ℝ²?
1. x² − y²
2. x² + y²
3. eˣ sin y
4. x³ − y³
Show answer
Answer: A — x² − y²; C — eˣ sin y
Δ(x² − y²) = 2 − 2 = 0 (it is Re z²). Δ(eˣ sin y) = eˣ sin y − eˣ sin y = 0 (it is Im eᶻ). Δ(x² + y²) = 4 ≠ 0. Δ(x³ − y³) = 6x − 6y, which is not identically 0.
If ∮_|z|=2 eᶻ/(z − 1)³ dz = kπi, where the circle is positively oriented, then k, correct to two decimal places, is ____.
Numerical answer — type the value.
Show answer
Answer: 2.72
z = 1 lies inside |z| = 2, and Cauchy’s formula for the second derivative gives ∮ eᶻ/(z − 1)³ dz = (2πi/2!)·e¹ = πie. So k = e = 2.718…, i.e. 2.72. Forgetting the 2! gives 2e.
The value of (1/π)∮_|z−i|=1 dz/(z² + 1), the circle positively oriented, is ____.
Numerical answer — type the value.
Show answer
Answer: 1
The poles are ±i; only i is inside |z − i| = 1 (−i is at distance 2). With g(z) = 1/(z + i), analytic inside, the integral is 2πi·g(i) = 2πi/(2i) = π, so the requested value is 1.
Which statements about entire functions are true?
1. a bounded entire function is constant
2. an entire function f with Re f(z) ≤ 5 for all z is constant
3. an entire function f with |f(z)| ≤ 1 + |z|² for all z is a polynomial of degree at most 2
4. sin z is bounded on ℂ
Show answer
Answer: A — a bounded entire function is constant; B — an entire function f with Re f(z) ≤ 5 for all z is constant; C — an entire function f with |f(z)| ≤ 1 + |z|² for all z is a polynomial of degree at most 2
(1) is Liouville. (2) |e^f| = e^{Re f} ≤ e⁵, so e^f is constant, hence f′e^f = 0 and f is constant. (3) Cauchy’s estimate on |z| = R gives |f⁽³⁾(0)| ≤ 3!(1 + R²)/R³ → 0, and likewise for higher derivatives at any point. (4) is false: |sin(iy)| = |sinh y| → ∞.
The maximum of |z² + 2z| over the closed disc |z| ≤ 1 is ____.
Numerical answer — type the value.
Show answer
Answer: 3
By the maximum modulus principle the maximum is on |z| = 1, where |z² + 2z| = |z||z + 2| = |z + 2| ≤ 3, with equality at z = 1. The value 3 = |1 + 2| is attained, so the maximum is 3.
Which statements are true?
1. there is an analytic f on the unit disc with f(1/n) = 1/n² for every n ≥ 2
2. there is an analytic f on the unit disc with f(1/n) = (−1)ⁿ/n for every n ≥ 2
3. an entire function that vanishes at every real number is identically zero
4. an analytic function on the unit disc with infinitely many zeros there is identically zero
Show answer
Answer: A — there is an analytic f on the unit disc with f(1/n) = 1/n² for every n ≥ 2; C — an entire function that vanishes at every real number is identically zero
(1) f(z) = z². (2) Even n force f = z near 0 and odd n force f = −z, by the identity theorem; contradiction. (3) ℝ has limit points in ℂ. (4) is false: sin(π/(1 − z)) vanishes at 1 − 1/k for every k ≥ 1, points accumulating only at the boundary point 1.
Which statements are true? (All circles are positively oriented.)
1. if f is continuous on a domain D and ∮_∂T f dz = 0 for every triangle T in D, then f is analytic on D
2. ∮_|z|=1 dz/z² = 0
3. ∮_|z|=1 z̄ dz = 0
4. if ∮_C f dz = 0 for one closed curve C, then f is analytic inside C
Show answer
Answer: A — if f is continuous on a domain D and ∮_{∂T} f dz = 0 for every triangle T in D, then f is analytic on D; B — ∮_{|z|=1} dz/z² = 0
(1) is Morera’s theorem. (2) 1/z² has the primitive −1/z on ℂ ∖ {0}, so its integral over any closed curve there is 0 (its residue is 0). (3) On |z| = 1, z̄ = 1/z, so the integral is 2πi. (4) is false: |z|² equals 1 on |z| = 1, so its integral there is ∮dz = 0, yet |z|² is analytic nowhere.
The radius of convergence of the Taylor series of f(z) = 1/(z² + 2z + 2) about z = 0, correct to two decimal places, is ____.
Numerical answer — type the value.
Show answer
Answer: 1.41
The singularities are the roots of z² + 2z + 2 = 0, z = −1 ± i, both at distance √2 from 0. f is analytic on |z| < √2 and not beyond, so R = √2 = 1.414, i.e. 1.41.
The radius of convergence of the Maclaurin series of tan z is
1. π/2
2. π
3. 1
4. ∞
Show answer
Answer: A — π/2
tan z = sin z/cos z is analytic except where cos z = 0, i.e. z = π/2 + kπ; these are all real, and the nearest to 0 are ±π/2. So R = π/2. The series having only odd powers does not change R.
In the Laurent expansion of f(z) = 1/((z − 1)(z − 2)) valid in 1 < |z| < 2, the coefficient of z² is ____.
Numerical answer — type the value.
Show answer
Answer: -0.125
f = 1/(z − 2) − 1/(z − 1). For |z| < 2, 1/(z − 2) = −(1/2)·1/(1 − z/2) = −Σ zⁿ/2ⁿ⁺¹; for |z| > 1, −1/(z − 1) = −Σ_n≥0 z⁻ⁿ⁻¹ has only negative powers. So the z² coefficient is −1/2³ = −0.125. In |z| < 1 the answer would be −1/8 + 1 = 0.875.