TRB PG Assistant - Analytic functions - limits, continuity, polynomials and rational functions, power series, Maclaurin series, uniform convergence, Abel's limit theorem Question Bank with Options

1. Which condition implies that f(z) = u(x, y) + iv(x, y) is analytic?

A) u and v are continuous and satisfy Cauchy-Riemann equations.

B) u and v are differentiable and satisfy Cauchy-Riemann equations.

C) u and v are harmonic and satisfy Cauchy-Riemann equations.

D) u and v are continuously differentiable and satisfy Cauchy-Riemann equations.

2. If a power series Σ aₙ (z-z₀)ⁿ has radius of convergence R, what can be said about the series at |z-z₀| = R?

A) It always converges.

B) It always diverges.

C) It may converge or diverge.

D) It converges if R is finite.

3. The theorem that states if f'(z) exists in a region, then f is analytic in that region is:

A) Cauchy-Riemann Theorem

B) Morera's Theorem

C) Liouville's Theorem

D) Goursat's Theorem

4. If f(z) = u(x,y) + iv(x,y) is analytic, and u(x,y) = x² - y², what is v(x,y)?

A) 2xy

B) -2xy

C) x² + y²

D) y² - x²

5. Let f(z) = Σ aₙ (z-z₀)ⁿ be a power series with radius of convergence R > 0. Then f(z) is:

A) Continuous in the disk |z-z₀| < R.

B) Differentiable in the disk |z-z₀| < R.

C) Analytic in the disk |z-z₀| < R.

D) All of the above.

6. What is the radius of convergence of the Maclaurin series for f(z) = 1/(1-z)?

A) 0

B) 1

C) ∞

D) 2

7. If f(z) = x + iy, is f(z) analytic?

A) Yes, because it is continuous.

B) No, because it does not satisfy Cauchy-Riemann equations.

C) Yes, because f'(z) = 1.

D) No, because it is not a polynomial.

8. Consider the power series Σ zⁿ/n. It converges for |z| < 1. What is its behavior at z=1?

A) It converges to -log(0).

B) It converges to -log(1-1).

C) It diverges.

D) It converges to log(2).

9. Which of the following is a necessary condition for a function to be analytic in a region?

A) It must be continuous.

B) It must be differentiable at every point in the region.

C) It must satisfy the Cauchy-Riemann equations at every point in the region.

D) Its real and imaginary parts must be harmonic.

10. If a sequence of functions {fₙ(z)} converges uniformly to f(z) on a compact set K, and each fₙ is analytic on K, then f(z) is:

A) Continuous on K

B) Analytic on K

C) Differentiable on K

D) Harmonic on K

11. What is the radius of convergence of the power series Σ (zⁿ / n!)?

A) 0

B) 1

C) ∞

D) e

12. Let f(z) = u(x,y) + iv(x,y). If f(z) is analytic, then u and v are called:

A) Harmonic conjugates

B) Cauchy-Riemann pairs

C) Analytic partners

D) Complex conjugates

13. Consider f(z) = z². Is it analytic at z=0?

A) No, because it's a polynomial.

B) Yes, because its partial derivatives satisfy the Cauchy-Riemann equations.

C) No, because it's not defined everywhere.

D) Yes, but only in a neighborhood of z=0.

14. What is the Maclaurin series for the function f(z) = 1/(1-z)?

A) 1 + z + z² + z³ + ...

B) 1 - z + z² - z³ + ...

C) z + z²/2 + z³/3 + ...

D) 1 + z²/2! + z⁴/4! + ...

15. According to Abel's Limit Theorem, if Σ aₙ converges, then what is the limit of Σ aₙ zⁿ as z approaches 1 from the left (z → 1⁻)?

A) 0

B) 1

C) Σ aₙ

D) ∞

16. If {fₙ(z)} converges uniformly to f(z) on S, and each fₙ is continuous on S, then f(z) is:

A) Continuous

B) Differentiable

C) Analytic

D) Harmonic

17. Let fₙ(z) = z/n for n = 1, 2, 3, ... . This sequence converges uniformly to f(z) = 0 on any bounded set.

A) True, for any bounded set.

B) False, only on compact sets.

C) False, it converges pointwise but not uniformly.

D) True, but only on the real axis.

18. If a power series converges to f(z) in a disk |z - z₀| < R, then f(z) is analytic in that disk. This is a consequence of:

A) Cauchy's Integral Theorem

B) Morera's Theorem

C) The definition of analyticity

D) The properties of uniform convergence

19. Consider the power series Σ n! zⁿ. What is its radius of convergence?

A) ∞

B) 1

C) 0

D) e

20. If f(z) = u(x, y) + iv(x, y) is analytic, and v(x, y) = x² + y², what is u(x, y)?

A) 2xy

B) -2xy

C) x² - y²

D) y² - x²

21. What is the domain of analyticity for the function f(z) = 1 / (z² + 1)?

A) All complex numbers.

B) All complex numbers except z = i and z = -i.

C) All complex numbers except z = 1 and z = -1.

D) All complex numbers except z = 0.

22. The function f(z) = Re(z) is:

A) Analytic everywhere.

B) Analytic only at z=0.

C) Not analytic anywhere.

D) Continuous everywhere.

23. Let f(z) = e^z. This function is analytic. What is its Maclaurin series?

A) 1 + z + z²/2! + z³/3! + ...

B) 1 - z + z²/2! - z³/3! + ...

C) z + z³/3! + z⁵/5! + ...

D) 1 + z²/2! + z⁴/4! + ...

24. If f(z) = x² - y² + i(2xy), what is f'(z)?

A) 2x + 2iy

B) 2z

C) 2x - 2iy

D) 2x

25. Cauchy's Integral Formula states that f(z₀) = (1/2πi) ∫γ f(z) / (z - z₀) dz, where γ is a simple closed contour enclosing z₀. This implies:

A) The value of an analytic function inside a contour is determined by its values on the contour.

B) The integral of an analytic function over a closed contour is always zero.

C) An analytic function is constant if its value at one point is zero.

D) An analytic function has a removable singularity at z₀.

26. If f(z) is analytic in a simply connected domain D, and γ is a simple closed contour in D, then ∫γ f(z) dz = ?

A) 2πi * f(z₀)

B) 0

C) f'(z₀)

D) 2πi

27. Let f(z) = Σ (z/2)ⁿ. This is a geometric series. For what values of z does it converge?

A) |z| < 2

B) |z| > 2

C) |z| = 2

D) All complex numbers.

28. Consider the power series Σ zⁿ. It converges for |z| < 1. What happens at z = 1?

A) It converges to 1/(1-z).

B) It converges to 1/(1-1) which is undefined.

C) It diverges.

D) It converges to 0.

29. Abel's Limit Theorem states that if a power series Σ aₙ zⁿ converges at a point z₀ on its circle of convergence, then:

A) The series converges uniformly in the region |z| ≤ |z₀|.

B) The function f(z) = Σ aₙ zⁿ is continuous at z₀.

C) The series converges absolutely at z₀.

D) The series converges to 0 at z₀.

30. Which theorem relates the uniform convergence of a sequence of functions to the continuity of the limit function?

A) Cauchy's Integral Theorem

B) Weierstrass M-Test

C) Abel's Limit Theorem

D) Uniform Convergence Theorem

31. If a sequence of analytic functions {fₙ(z)} converges uniformly to f(z) on a region R, then the sequence of their derivatives {fₙ'(z)}:

A) Converges uniformly to f'(z) on R.

B) Converges uniformly to some function on R.

C) May not converge at all.

D) Converges uniformly to f(z).

32. If a sequence of analytic functions {fₙ(z)} converges uniformly to f(z) on a region R, then f(z) is:

A) Continuous on R

B) Analytic on R

C) Differentiable on R

D) Harmonic on R

33. Uniform convergence of a sequence of functions {fₙ(z)} to f(z) on a set S means:

A) For every ε > 0, there exists δ > 0 such that if |z - z₀| < δ, then |fₙ(z) - f(z)| < ε for all n.

B) For every ε > 0, there exists N such that if n > N, then |fₙ(z) - f(z)| < ε for all z in S.

C) For every ε > 0, there exists N such that if n > N, then |fₙ(z) - f(z)| < ε for some z in S.

D) For every ε > 0, there exists δ > 0 such that if |fₙ(z) - f(z)| < ε, then |z - z₀| < δ.

34. The Maclaurin series for sin(z) is:

A) z - z³/3! + z⁵/5! - ...

B) 1 - z²/2! + z⁴/4! - ...

C) z + z²/2! + z³/3! + ...

D) 1 + z + z²/2! + ...

35. The Maclaurin series for e^z is given by:

A) Σ (zⁿ / n!) from n=0 to ∞

B) Σ (zⁿ / n) from n=1 to ∞

C) Σ (-1)ⁿ zⁿ from n=0 to ∞

D) Σ n! zⁿ from n=0 to ∞

36. What is the Maclaurin series of a function f(z)?

A) A Taylor series centered at z = 1.

B) A Taylor series centered at z = 0.

C) A power series with only negative powers of z.

D) A Laurent series with only negative powers of z.

37. If a power series converges for |z - z₀| < R and diverges for |z - z₀| > R, what is R called?

A) Radius of divergence

B) Radius of convergence

C) Radius of analyticity

D) Radius of continuity

38. What is the radius of convergence of a power series Σ aₙ (z - z₀)ⁿ?

A) The largest radius R such that the series converges for |z - z₀| < R.

B) The smallest radius R such that the series converges for |z - z₀| > R.

C) The value of R for which the series converges at |z - z₀| = R.

D) The value of R for which the series diverges at |z - z₀| = R.

39. What is a power series centered at z₀?

A) A series of the form Σ aₙ (z - z₀)ⁿ

B) A series of the form Σ aₙ zⁿ

C) A series of the form Σ aₙ / (z - z₀)ⁿ

D) A series of the form Σ aₙ z⁻ⁿ

40. Consider the rational function R(z) = 1/z. At which point is it not analytic?

A) z = 1

B) z = i

C) z = 0

D) z = -1

41. A rational function is defined as the ratio of two polynomials. Is every rational function analytic?

A) Yes, everywhere.

B) No, it is not analytic at the roots of the denominator polynomial.

C) Yes, except at z=0.

D) No, only polynomials are analytic.

42. Which of the following is NOT a polynomial in z?

A) P(z) = 3z² - 2z + 5

B) P(z) = z⁵

C) P(z) = z + 1/z

D) P(z) = 7

43. If f(z) = u(x, y) + iv(x, y) is analytic, and f'(z) = 0 in a region, what can be said about f(z)?

A) f(z) is identically zero.

B) f(z) is a non-zero constant.

C) f(z) is a non-constant function.

D) f(z) is purely imaginary.

44. What is the derivative of f(z) = z³ at z = 1 + i?

A) 3(1+i)²

B) 3(1+i)

C) 3(1+i)³

D) 2(1+i)

45. Let f(z) = |z|². Is this function analytic?

A) Yes, it is analytic everywhere.

B) No, it is not differentiable at z=0.

C) Yes, it is analytic only at z=0.

D) No, it is not continuous everywhere.

46. Consider the function f(z) = z². Is this function analytic?

A) No, because it's a polynomial.

B) Yes, because it is differentiable everywhere.

C) No, because it is not defined for all complex numbers.

D) Yes, but only at z=0.

47. Which theorem states that if a function is analytic in a region, its real and imaginary parts satisfy Laplace's equation?

A) Cauchy-Riemann Theorem

B) Liouville's Theorem

C) Mean Value Theorem for Harmonic Functions

D) Harmonic Conjugate Theorem

48. If a function f(z) = u(x, y) + iv(x, y) is analytic in a region, what conditions must its real and imaginary parts satisfy?

A) They must be continuous everywhere.

B) They must satisfy the Cauchy-Riemann equations.

C) They must be harmonic.

D) They must be differentiable in the region.

49. What is the fundamental condition for a complex function f(z) to be analytic at a point z₀?

A) The function must be continuous at z₀.

B) The function must be differentiable in a neighborhood of z₀.

C) The Cauchy-Riemann equations must be satisfied at z₀.

D) The partial derivatives of the real and imaginary parts must exist at z₀.

Analytic functions - limits, continuity, polynomials and rational functions, power series, Maclaurin series, uniform convergence, Abel's limit theorem - Question Bank