Zeros and poles, local mapping properties, maximum modulus principle - Question Bank

1. What is the order of the pole at z = 0 for f(z) = z^-1 * cot(z)?
A) 1
B) 2
C) 3
D) 4
2. Let f(z) = (z^2 - 1) / (z - 1). What is the nature of the singularity at z = 1?
A) Pole of order 1
B) Removable singularity
C) Essential singularity
D) Branch point
3. If f(z) is analytic and non-constant in the open disk |z| < R, and f(z) is never zero in this disk, then |f(z)| attains its minimum value:
A) At the center z=0
B) On the boundary |z|=R
C) At an interior point where f'(z) = 0
D) This scenario is impossible
4. What is the order of the zero at z=0 for f(z) = e^z - 1?
A) 0
B) 1
C) 2
D) infinity
5. If f(z) is analytic in a domain D, and z0 is an interior point of D, what is the implication of f(z0) = 0 for the Maximum Modulus Principle?
A) It implies |f(z)| attains its maximum at z0
B) It implies |f(z)| attains its minimum at z0
C) It implies f(z) is constant
D) It implies f'(z0) is non-zero
6. Consider the function f(z) = 1/(z-a). What is the order of the pole at z=a?
A) 0
B) 1
C) 2
D) infinity
7. What is the order of the pole at z=0 for f(z) = z^-2 * cos(z)?
A) 0
B) 1
C) 2
D) 3
8. If f(z) is analytic in the domain |z| < 1 and |f(z)| <= 1 for all z in the domain, and |f(z_0)| = 1 for some z_0 with |z_0| < 1, what can be concluded?
A) f(z) is constant
B) f(z) must be zero at z_0
C) f(z) has a pole at z_0
D) f(z) must be identically 1
9. What is the order of the zero at z=0 for f(z) = sinh(z) - z?
A) 1
B) 2
C) 3
D) 4
10. According to the Maximum Modulus Principle, if a non-constant analytic function f(z) has a maximum modulus inside a domain D, then:
A) The maximum modulus must be zero
B) The maximum modulus must be 1
C) This is impossible
D) The function must be constant
11. If f(z) is analytic in a punctured disk 0 < |z-z0| < R and has a pole of order m at z0, then |f(z)| tends to ____ as z approaches z0.
A) 0
B) 1
C) infinity
D) A finite non-zero value
12. What is the order of the pole at z = 0 for f(z) = 1 / (z * (1 - cos(z)))?
A) 1
B) 2
C) 3
D) 4
13. Consider the function f(z) = (z-2)^3. What is the order of the zero at z=2?
A) 0
B) 1
C) 2
D) 3
14. If f(z) is analytic in the unit disk |z| <= 1 and |f(z)| = 1 on the boundary |z|=1, what can be concluded about f(z)?
A) f(z) is constant
B) f(z) has a zero inside the disk
C) f(z) has a pole inside the disk
D) f(z) is never zero inside the disk
15. What is the order of the pole at z = pi for f(z) = cot(z)?
A) 1
B) 2
C) 3
D) 4
16. If f(z) is analytic in a region and f(z_0) = f'(z_0) = ... = f^(m-1)(z_0) = 0 and f^(m)(z_0) != 0, then f(z) has a zero of order:
A) m-1
B) m
C) m+1
D) 0
17. What is the order of the pole at z = 0 for f(z) = z^-3 * e^z?
A) 1
B) 2
C) 3
D) 4
18. Consider f(z) = z^2. What is the order of the zero of f(z) at z=0?
A) 0
B) 1
C) 2
D) infinity
19. If f(z) is analytic and non-constant in a domain D, and f(z) != 0 in D, then |f(z)| attains its minimum value:
A) Inside D
B) On the boundary of D
C) At a point where f(z) = 0
D) At a pole
20. What is the order of the zero at z = 0 for the function f(z) = 1 - cos(z)?
A) 1
B) 2
C) 3
D) 4
21. If f(z) has a pole of order m at z0, then lim (z-z0)^m * f(z) as z->z0 is:
A) 0
B) 1
C) A finite non-zero number
D) Infinity
22. Let f(z) = z^n. For what value of n does f(z) have a zero of order 5 at z=0?
A) 0
B) 1
C) 5
D) infinity
23. What is the order of the pole at z = 0 for f(z) = 1 / (z^2 * sin(z))?
A) 1
B) 2
C) 3
D) 4
24. If f(z) is analytic in a domain D and has a zero of order m at z0, what is the order of the zero of f'(z) at z0?
A) m-1
B) m
C) m+1
D) 0
25. Consider f(z) = e^z. What is the maximum value of |f(z)| on the rectangle defined by 0 <= Re(z) <= 1 and 0 <= Im(z) <= pi?
A) e
B) e*pi
C) e^pi
D) e^(1+pi)
26. If f(z) is analytic in the punctured disk 0 < |z| < 1 and has a pole at z = 0, can |f(z)| attain a minimum value in this domain?
A) Yes, at z=0
B) Yes, at some point in the domain
C) No, unless f(z) is constant
D) No, because of the pole
27. The Minimum Modulus Principle states that if f(z) is analytic and non-constant in a bounded domain D and continuous on its boundary ∂D, and if f(z) is never zero in D, then the minimum value of |f(z)| occurs:
A) At an interior point of D
B) At a point where f(z) = 0
C) On the boundary ∂D
D) At a pole
28. What is the minimum value of |f(z)| for f(z) = z - 1 on the closed disk |z| <= 1?
A) 0
B) 1
C) 2
D) The minimum is not attained
29. If f(z) is analytic on the closed unit disk |z| <= 1 and f(0) = 0, what can be said about |f(z)| inside the disk?
A) |f(z)| = 0 for all z
B) |f(z)| > 0 for z != 0
C) |f(z)| < |f'(0)| for z != 0
D) |f(z)| attains its maximum at z=0
30. Consider the function f(z) = z^2. What is the maximum value of |f(z)| on the disk |z| <= 2?
A) 0
B) 2
C) 4
D) 8
31. If f(z) is analytic in a domain D and |f(z)| is constant on D, what can be concluded about f(z)?
A) f(z) is identically zero
B) f(z) is a non-zero constant
C) f(z) has zeros in D
D) f(z) has poles in D
32. What does the Maximum Modulus Principle imply for a non-constant analytic function f(z) inside a disk |z| <= R?
A) The maximum of |f(z)| occurs at the center
B) The maximum of |f(z)| occurs on the boundary |z| = R
C) The maximum of |f(z)| occurs at a removable singularity
D) The maximum of |f(z)| is attained at an interior point
33. According to the Maximum Modulus Principle, if f(z) is analytic and non-constant in a bounded domain D and continuous on its boundary ∂D, where does the maximum value of |f(z)| occur?
A) At an interior point of D
B) At a point where f(z) = 0
C) On the boundary ∂D
D) It does not attain a maximum
34. If f(z) is analytic and non-constant in a domain D, which principle states that |f(z)| cannot attain a maximum value in D?
A) Cauchy's Integral Theorem
B) Maximum Modulus Principle
C) Liouville's Theorem
D) Residue Theorem
35. What is the nature of the singularity at z = 0 for f(z) = sin(1/z)?
A) Removable singularity
B) Pole
C) Essential singularity
D) None of the above
36. Consider the function f(z) = z^3 / (sin(z) - z + z^3/6). What is the order of the zero at z = 0?
A) 1
B) 2
C) 3
D) 4
37. If f(z) = 1/z^2, what is the order of the pole at z = 0?
A) 1
B) 2
C) 3
D) 4
38. What is the order of the zero at z = 0 for f(z) = z * sin(z)?
A) 1
B) 2
C) 3
D) 4
39. If f(z) has a simple pole at z0, what is the residue of f(z) at z0?
A) The coefficient of (z-z0)^-1 in the Laurent series
B) The coefficient of (z-z0)^0 in the Laurent series
C) The coefficient of (z-z0)^1 in the Laurent series
D) Zero
40. What is the principal part of the Laurent series expansion of f(z) = 1 / (z * (z-1)) around z = 0?
A) 1/z
B) -1/z
C) 1/(z-1)
D) -1/(z-1)
41. If f(z) has a zero of order m at z0, then f'(z0) is:
A) Always zero
B) Never zero
C) Zero if m > 1
D) Zero if m = 1
42. What is the order of the pole at z = 1 for the function f(z) = 1 / (z-1)^3?
A) 1
B) 2
C) 3
D) 4
43. Let f(z) = (e^z - 1 - z) / z^2. What is the order of the zero at z = 0?
A) 1
B) 2
C) 3
D) 4
44. If f(z) is analytic in a region D and has a zero of order m at z0 in D, what can be said about f'(z) at z0?
A) f'(z0) = 0
B) f'(z0) is non-zero
C) f'(z0) is infinite
D) f'(z0) is undefined
45. What is the order of the pole at z = 0 for the function h(z) = cot(z)?
A) 1
B) 2
C) 3
D) 4
46. Consider the function f(z) = z^2 * sin(z). What is the order of the zero at z = 0?
A) 1
B) 2
C) 3
D) 4
47. If f(z) has a pole of order m at z0, then what is the behavior of 1/f(z) near z0?
A) 1/f(z) has a zero of order m at z0
B) 1/f(z) has a pole of order m at z0
C) 1/f(z) is analytic and non-zero at z0
D) 1/f(z) is constant near z0
48. What is the order of the pole at z = 0 for the function g(z) = 1/sin(z)?
A) 0
B) 1
C) 2
D) 3
49. If f(z) has a zero of order m at z0, then what is the behavior of f(z) near z0?
A) f(z) is analytic and non-zero at z0
B) f(z) has a pole of order m at z0
C) f(z) behaves like c(z-z0)^m for some non-zero constant c near z0
D) f(z) is constant near z0
50. What is the order of a zero at z = 0 for the function f(z) = sin(z) - z + z^3/6?
A) 1
B) 2
C) 3
D) 4
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