Cauchy Riemann Equations and Analytic Functions - Question Bank
1. If f(z) = u + iv is analytic, then f'(z) can be expressed using polar coordinates (r, θ) as:
2. Which form of the Cauchy-Riemann equations is used when representing f in terms of z and z̄?
3. If f(z) is analytic in a domain D, then f(z) is necessarily:
4. The function f(z) = |z| is:
5. For f(z) = e^(x+iy), what are u and v?
6. If f(z) = u + iv is analytic, and f'(z) = 2z, then f(z) is:
7. If f(z) = x² + i y², is f(z) analytic?
8. The converse of the Cauchy-Riemann equations (i.e., if CR equations hold, is the function analytic?) is true if:
9. 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?
10. If f(z) = u + iv is analytic and u(x, y) = x², what is v(x, y) up to an additive constant?
11. Which of the following is a consequence of the Cauchy-Riemann equations?
12. If f(z) = u + iv is analytic, then f'(z) = 0 implies that f(z) is:
13. The condition u_x = v_y and u_y = -v_x is known as the:
14. If f(z) is analytic in a simply connected domain D, then f'(z) is also:
15. The function f(z) = log(z) is analytic in the complex plane except for:
16. If f(z) = e^z, then f'(z) is:
17. If f(z) = sin(z), then f'(z) is:
18. If f(z) = cos(z), then f'(z) is:
19. 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:
20. Let f(z) = u + iv be analytic. If u(x, y) = ln(x² + y²), what is v(x, y) up to an additive constant?
21. If f(z) = z * |z|², is f(z) analytic?
22. The Cauchy-Riemann equations provide a necessary condition for a function to be analytic. What additional condition is needed?
23. 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?
24. Which of the following is an analytic function?
25. If f(z) = u + iv is analytic, then ∂²v/∂x² + ∂²v/∂y² equals:
26. The function f(z) = 1/z is analytic in its domain of definition. What is its domain?
27. If f(z) = z + 1/z, what is f'(z)?
28. The function f(z) = x + iy is:
29. If f(z) = u + iv is analytic, then f'(z) can also be written as:
30. If u(x, y) = e^x cos y, find its harmonic conjugate v(x, y) such that f(z) = u + iv is analytic.
31. If u(x, y) = x² - y², find its harmonic conjugate v(x, y) such that f(z) = u + iv is analytic.
32. The function v(x, y) is called the harmonic conjugate of u(x, y) if:
33. If f(z) is analytic, the function u(x, y) is said to be:
34. If f(z) = u(x, y) + iv(x, y) is analytic, then ∂²u/∂x² + ∂²u/∂y² equals:
35. The function f(z) = Im(z) = y is:
36. The function f(z) = Re(z) = x is:
37. Consider f(z) = e^x (cos y + i sin y). Which of the following is true?
38. If f(z) is analytic in a domain D, then f'(z) exists at every point in D and is given by:
39. The function f(z) = |z|² is:
40. For f(z) = x³ - 3xy² + i(3x²y - y³), what are ∂u/∂y and -∂v/∂x?
41. For f(z) = x³ - 3xy² + i(3x²y - y³), what are ∂u/∂x and ∂v/∂y?
42. Which of the following functions is NOT analytic at z = 0?
43. The condition that the partial derivatives of u and v are continuous is:
44. If f(z) is analytic, then its real and imaginary parts, u and v, are:
45. A function f(z) is analytic in a domain D if it is:
46. For f(z) = z², what are the components u(x, y) and v(x, y)?
47. If f(z) = z², what are the Cauchy-Riemann equations satisfied by u(x, y) and v(x, y)?
48. The Cauchy-Riemann equations for a function f(z) = u(x, y) + iv(x, y) are given by:
49. If a function f(z) = u(x, y) + iv(x, y) is analytic, which pair of partial differential equations must be satisfied?
50. What is the fundamental condition for a complex function f(z) = u(x, y) + iv(x, y) to be analytic in a region?