Complex Analysis

JKPSC Assistant Professor (Mathematics) · Mathematics · 48 practice questions with answers and explanations: 25 Easy, 21 Medium, 2 Hard.

Sample questions

Sample Easy question 1

The value of i⁴ is:

  1. −1
  2. 1
  3. −i
  4. i
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Answer: B. i has period 4: i⁴ = i⁰ = 1.

Sample Easy question 2

What is the sum of all the 8th roots of unity?

  1. −1
  2. 8
  3. 1
  4. 0
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Answer: D. The roots are the vertices of a regular polygon centred at the origin, so they sum to 0.

Sample Easy question 3

The conjugate of 1 + 4i is:

  1. 1 − 4i
  2. −1 − 4i
  3. −1 + 4i
  4. 4 + 1i
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Answer: A. Change the sign of the imaginary part.

Sample Medium question 4

Schwarz's lemma applies to analytic maps of the unit disc into itself that fix:

  1. The origin
  2. The point 1
  3. The boundary
  4. Every point
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Answer: A. Then |f(z)| ≤ |z| and |f′(0)| ≤ 1.

Sample Medium question 5

Find the product (2 − 2i)(−2 + 1i).

  1. −2 + 6i
  2. −2 − 2i
  3. −2 − 6i
  4. −6 − 2i
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Answer: A. Multiply out and use i² = −1.

Sample Medium question 6

The Cauchy–Riemann equations for f = u + iv are:

  1. u_x = u_y and v_x = v_y
  2. u_x = v_y and u_y = −v_x
  3. u_x = −v_y and u_y = v_x
  4. u_x = v_x and u_y = v_y
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Answer: B. These conditions are necessary for complex differentiability.

Sample Hard question 7

The residue of e^z ÷ z² at z = 0 is:

  1. 1
  2. 2
  3. e
  4. 0
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Answer: A. e^z ÷ z² = 1 ÷ z² + 1 ÷ z + ½ + …

Sample Hard question 8

The radius of convergence of the Taylor series of 1 ÷ (1 + z²) about z = 0 is:

  1. 2
  2. ∞
  3. 1
  4. π
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Answer: C. The nearest singularities are at z = ±i.

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