TRB BT Assistant - Cauchy Riemann Equations and Analytic Functions Online Test (MCQ Quiz)

1. What is the fundamental condition for a complex function f(z) = u(x, y) + iv(x, y) to be analytic in a region?

A) The partial derivatives of u and v are continuous and satisfy the Cauchy-Riemann equations. B) The function f(z) is continuous and differentiable at every point in the region. C) The real part u(x, y) and imaginary part v(x, y) are harmonic functions. D) The function f(z) can be represented by a Taylor series in the region.

2. If a function f(z) = u(x, y) + iv(x, y) is analytic, which pair of partial differential equations must be satisfied?

A) ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x B) ∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x C) ∂u/∂x = ∂v/∂x and ∂u/∂y = ∂v/∂y D) ∂u/∂x = -∂v/∂x and ∂u/∂y = -∂v/∂y

3. The Cauchy-Riemann equations for a function f(z) = u(x, y) + iv(x, y) are given by:

A) u_x = v_y, u_y = -v_x B) u_x = -v_y, u_y = v_x C) u_x = v_x, u_y = v_y D) u_x = -v_x, u_y = -v_y

4. If f(z) = z², what are the Cauchy-Riemann equations satisfied by u(x, y) and v(x, y)?

A) 2x = 2y and 2y = -2x B) 2x = -2y and 2y = 2x C) 2x = 2x and 2y = 2y D) x = y and y = -x

5. For f(z) = z², what are the components u(x, y) and v(x, y)?

A) u = x² - y², v = 2xy B) u = x² + y², v = 2xy C) u = 2xy, v = x² - y² D) u = x², v = y²

6. A function f(z) is analytic in a domain D if it is:

A) Differentiable at every point in D. B) Continuous at every point in D. C) Harmonic at every point in D. D) Bounded at every point in D.

7. If f(z) is analytic, then its real and imaginary parts, u and v, are:

A) Harmonic functions. B) Analytic functions themselves. C) Entire functions. D) Meromorphic functions.

8. The condition that the partial derivatives of u and v are continuous is:

A) Necessary and sufficient for analyticity. B) Sufficient but not necessary for analyticity. C) Necessary but not sufficient for analyticity. D) Neither necessary nor sufficient for analyticity.

9. Which of the following functions is NOT analytic at z = 0?

A) f(z) = z³ B) f(z) = e^z C) f(z) = sin(z) D) f(z) = 1/z

10. For f(z) = x³ - 3xy² + i(3x²y - y³), what are ∂u/∂x and ∂v/∂y?

A) ∂u/∂x = 3x² - 3y², ∂v/∂y = 3x² - 3y² B) ∂u/∂x = 3x² - 3y², ∂v/∂y = 3y² - 3x² C) ∂u/∂x = 3x² + 3y², ∂v/∂y = 3x² + 3y² D) ∂u/∂x = x³ - 3y², ∂v/∂y = x² - y³

11. For f(z) = x³ - 3xy² + i(3x²y - y³), what are ∂u/∂y and -∂v/∂x?

A) ∂u/∂y = -6xy, -∂v/∂x = -(6xy) = -6xy B) ∂u/∂y = 6xy, -∂v/∂x = -(6xy) = -6xy C) ∂u/∂y = -6xy, -∂v/∂x = 6xy D) ∂u/∂y = 3y², -∂v/∂x = -3x²

12. The function f(z) = |z|² is:

A) Differentiable only at z = 0. B) Analytic everywhere. C) Continuous everywhere but not differentiable. D) Not continuous at z = 0.

13. If f(z) is analytic in a domain D, then f'(z) exists at every point in D and is given by:

A) f'(z) = ∂u/∂x + i ∂v/∂x B) f'(z) = ∂u/∂y + i ∂v/∂y C) f'(z) = ∂u/∂x - i ∂u/∂y D) f'(z) = ∂v/∂y - i ∂v/∂x

14. Consider f(z) = e^x (cos y + i sin y). Which of the following is true?

A) f(z) is analytic and f'(z) = e^z. B) f(z) is analytic and f'(z) = e^x (cos y + i sin y). C) f(z) is not analytic. D) f(z) satisfies the Cauchy-Riemann equations only at z = 0.

15. The function f(z) = Re(z) = x is:

A) Not analytic anywhere. B) Analytic everywhere. C) Analytic only at z = 0. D) Differentiable everywhere but not analytic.

16. The function f(z) = Im(z) = y is:

A) Not analytic anywhere. B) Analytic everywhere. C) Analytic only at z = 0. D) Differentiable everywhere but not analytic.

17. If f(z) = u(x, y) + iv(x, y) is analytic, then ∂²u/∂x² + ∂²u/∂y² equals:

A) 0 B) 1 C) u D) v

18. If f(z) is analytic, the function u(x, y) is said to be:

A) Harmonic. B) Analytic. C) Holomorphic. D) Entire.

19. The function v(x, y) is called the harmonic conjugate of u(x, y) if:

A) f(z) = u + iv is analytic. B) u and v are harmonic. C) u and v satisfy the Cauchy-Riemann equations. D) f(z) is differentiable.

20. If u(x, y) = x² - y², find its harmonic conjugate v(x, y) such that f(z) = u + iv is analytic.

A) v = 2xy B) v = -2xy C) v = x² + y² D) v = y² - x²

21. If u(x, y) = e^x cos y, find its harmonic conjugate v(x, y) such that f(z) = u + iv is analytic.

A) v = e^x sin y B) v = -e^x sin y C) v = e^y cos x D) v = e^y sin x

22. If f(z) = u + iv is analytic, then f'(z) can also be written as:

A) f'(z) = ∂v/∂y - i ∂u/∂y B) f'(z) = ∂v/∂x - i ∂u/∂x C) f'(z) = ∂u/∂y + i ∂v/∂y D) f'(z) = ∂u/∂x + i ∂u/∂y

23. The function f(z) = x + iy is:

A) Analytic everywhere. B) Not analytic anywhere. C) Analytic only at z = 0. D) Differentiable everywhere but not analytic.

24. If f(z) = z + 1/z, what is f'(z)?

A) 1 - 1/z² B) 1 + 1/z² C) 1 - 1/z D) 1 + 1/z

25. The function f(z) = 1/z is analytic in its domain of definition. What is its domain?

A) All complex numbers except z = 0. B) All complex numbers. C) The open disk |z| < 1. D) The closed disk |z| >= 1.

26. If f(z) = u + iv is analytic, then ∂²v/∂x² + ∂²v/∂y² equals:

A) 0 B) 1 C) u D) v

27. Which of the following is an analytic function?

A) f(z) = z̄ B) f(z) = Re(z) C) f(z) = Im(z) D) f(z) = z³

28. If f(z) = u(x, y) + iv(x, y) is analytic, and u(x, y) = x² - y², then what is v(x, y) up to an additive constant?

A) 2xy B) -2xy C) x² + y² D) y² - x²

29. The Cauchy-Riemann equations provide a necessary condition for a function to be analytic. What additional condition is needed?

A) The continuity of the first partial derivatives. B) The continuity of the function itself. C) The harmonic nature of the real and imaginary parts. D) The existence of the second partial derivatives.

30. If f(z) = z * |z|², is f(z) analytic?

A) No, only differentiable at z=0. B) Yes, analytic everywhere. C) No, not even differentiable at z=0. D) Yes, analytic except at z=0.

31. Let f(z) = u + iv be analytic. If u(x, y) = ln(x² + y²), what is v(x, y) up to an additive constant?

A) -tan⁻¹(y/x) B) tan⁻¹(y/x) C) ln(x² + y²) D) -ln(x² + y²)

32. The derivative of an analytic function f(z) = u + iv at a point z₀ can be computed using only the partial derivatives of u and v at z₀ as:

A) ∂u/∂x + i ∂v/∂x B) ∂u/∂y + i ∂v/∂y C) ∂v/∂x + i ∂u/∂x D) ∂v/∂y + i ∂u/∂y

33. If f(z) = cos(z), then f'(z) is:

A) -sin(z) B) sin(z) C) cos(z) D) -cos(z)

34. If f(z) = sin(z), then f'(z) is:

A) cos(z) B) -cos(z) C) sin(z) D) -sin(z)

35. If f(z) = e^z, then f'(z) is:

A) e^z B) e^x C) e^y D) ze^(z-1)

36. The function f(z) = log(z) is analytic in the complex plane except for:

A) The non-positive real axis. B) The positive real axis. C) The imaginary axis. D) The origin.

37. If f(z) is analytic in a simply connected domain D, then f'(z) is also:

A) Analytic in D. B) Continuous in D. C) Differentiable in D. D) Harmonic in D.

38. The condition u_x = v_y and u_y = -v_x is known as the:

A) Cauchy-Riemann equations. B) Laplace equation. C) Poincare equation. D) Euler-Lagrange equation.

39. If f(z) = u + iv is analytic, then f'(z) = 0 implies that f(z) is:

A) A constant function. B) An odd function. C) An even function. D) A purely imaginary function.

40. Which of the following is a consequence of the Cauchy-Riemann equations?

A) The real and imaginary parts are harmonic. B) The function is always real-valued. C) The function is always purely imaginary. D) The function is constant.

41. If f(z) = u + iv is analytic and u(x, y) = x², what is v(x, y) up to an additive constant?

A) -2xy B) 2xy C) x² D) y²

42. If f(z) = u + iv is analytic and v(x, y) = e^x sin y, what is u(x, y) up to an additive constant?

A) e^x cos y B) -e^x cos y C) e^y sin x D) e^y cos x

43. The converse of the Cauchy-Riemann equations (i.e., if CR equations hold, is the function analytic?) is true if:

A) The first partial derivatives of u and v are continuous. B) The function is continuous. C) The function is differentiable. D) The second partial derivatives exist.

44. If f(z) = x² + i y², is f(z) analytic?

A) No, it fails Cauchy-Riemann equations. B) Yes, it is analytic everywhere. C) No, it is only differentiable at z=0. D) Yes, it is analytic except at z=0.

45. If f(z) = u + iv is analytic, and f'(z) = 2z, then f(z) is:

A) z² + C B) z² C) 2z + C D) z + C

46. For f(z) = e^(x+iy), what are u and v?

A) u = e^x cos y, v = e^x sin y B) u = e^x sin y, v = e^x cos y C) u = e^x, v = e^y D) u = cos y, v = sin y

47. The function f(z) = |z| is:

A) Not analytic anywhere. B) Analytic everywhere. C) Differentiable only at z=0. D) Continuous everywhere.

48. If f(z) is analytic in a domain D, then f(z) is necessarily:

A) Continuous in D. B) Differentiable in D. C) Harmonic in D. D) Bounded in D.

49. Which form of the Cauchy-Riemann equations is used when representing f in terms of z and z̄?

A) ∂f/∂z̄ = 0 B) ∂f/∂z = 0 C) ∂f/∂x = 0 D) ∂f/∂y = 0

50. If f(z) = u + iv is analytic, then f'(z) can be expressed using polar coordinates (r, θ) as:

A) e^(-iθ) (∂u/∂r + i ∂v/∂r) B) e^(iθ) (∂u/∂r + i ∂v/∂r) C) ∂u/∂r + i ∂v/∂r D) ∂u/∂θ + i ∂v/∂θ

Cauchy Riemann Equations and Analytic Functions - Online Test

Test Results