Complex Analysis - Quiz App

Q1: Let f(z) be an entire function such that |f(z)| ≤ e^{|z|} for all z ∈ C. Which of the following is TRUE?

f(z) is constant f(z) is a polynomial of degree at most 1 f(z) is of the form e^{az} where |a| ≤ 1 f(z) can be any entire function with order ≤ 1

Q2: Which of the following is TRUE about the function f(z) = 1/(z² + 1) at z = i?

z = i is a simple pole z = i is a removable singularity z = i is an essential singularity z = i is a pole of order 2

Q3: Let f(z) be analytic in a domain D containing the unit circle. If ∫_{|z|=1} f(z) dz = 0, which of the following is necessarily TRUE?

f(z) is analytic inside the unit circle f(z) has no singularities inside the unit circle f(z) is constant f(z) is entire

Q4: For the function f(z) = z³ sin(1/z), which of the following is TRUE about z = 0?

z = 0 is a removable singularity z = 0 is a pole of order 2 z = 0 is an essential singularity z = 0 is a zero of order 4

Q5: Let f(z) = e^{1/z}. What is the residue of f(z) at z = 0?

0 1 ∞ The residue does not exist

Q6: If f(z) = u(x,y) + iv(x,y) is analytic, which of the following is ALWAYS TRUE?

u and v satisfy Laplace's equation u is harmonic but v is not necessarily harmonic v is harmonic but u is not necessarily harmonic Neither u nor v need be harmonic

Q7: What is the radius of convergence of the power series Σ(n!)² z^n?

0 1 ∞ 1/e²

Q8: Which of the following is NOT a conformal mapping from the upper half-plane to the unit disk?

w = (z - i)/(z + i) w = (z - a)/(z - ar{a}) where Im(a) > 0 w = e^{iz} w = (z - i)/(z + i) with appropriate scaling

Q9: For the function f(z) = Log(z) (principal branch), where is f(z) analytic?

C {0} C ( -∞, 0] C [0, ∞) C {z : Re(z) ≤ 0 and Im(z) = 0}

Q10: Let f(z) = z/(z² - 1). What is the residue at z = 1?

1/2 -1/2 1 -1

Q11: Which of the following statements about the function f(z) = sin(z)/z is TRUE?

z = 0 is a removable singularity z = 0 is a simple pole z = 0 is an essential singularity z = 0 is a pole of order 2

Q12: What is the value of ∫_{|z|=2} dz/(z² + 1)?

0 πi 2πi 4πi

Q13: Let f(z) = z² + 1. Which of the following is TRUE about the function?

f(z) is entire and maps R to R f(z) is entire but not conformal anywhere f(z) is entire and conformal everywhere f(z) is entire but its derivative vanishes at z = 0

Q14: For the Laurent series expansion of f(z) = 1/(z²(1-z)) in the annulus 0 < |z| < 1, what is the coefficient of 1/z²?

1 -1 0 2

Q15: Which of the following functions has an essential singularity at z = 0?

sin(1/z) 1/z² 1/(z-1) Log(z)

Q16: Let f(z) be analytic in the disk |z| < 1 and |f(z)| ≤ 1 for all |z| < 1. If f(0) = 0, which of the following is TRUE?

|f(z)| ≤ |z| for all |z| < 1 |f(z)| ≤ |z|² for all |z| < 1 f(z) = e^{iθ}z for some θ f(z) is constant