Complex Analysis Answers

 Given that f(z) is holomorphic everywhere except at z=−1, the value of ∂v/∂y−∂v/∂x at the point (1,1) can be written as a/b where a and b are coprime positive integers. What is 100a+b?

For two real numbers x and y, f(x+iy) can be written as u(x,y)+iv(x,y) where u and v are functions of real variables.

 What is ((x+1)^2+y^2)u(x,y)?

a) (x+1)^2+y^2

b) x^2+y^2−1

c) y^2−x^2+1

d) (x−y)^2

A fixed point of f(z) is a complex number z such that f(z)=z. How many fixed points does f(z) have?

Let α be a complex number that satisfies both the equations from the previous question. Which of the following are possible arguments of the cube roots of α?

a) π/6

b) 5π/6

c) π/12

d) π/2

e) 5π/12

f) 0

g) π

h) -π/2

Let f(z)=z−1/z+1

. The questions that follow will all revolve around this complex-valued function.

1.

 The value of 25f(2+4i) is equal to a+ib where a,b∈R, and i^2=−1. What is the value of 10a+b?

Let w=f(z)=z^2+6+6i is a complex function. Then the image of the line x=5

 looks like what?