Complex Analysis Answers

find the residue of f(z)=z^2+2z/(z+1)^2(z+4)at its poles

z₁=3.45∠980°

z₂₌-5+9i

1. z1+z2 final answer in polar form
2. z1-z2 final answer in trigonometric form
3. z2*z1 final answer in exponential form
4. z1/z2 final answer in rectangular form

1a) Find the real numbers a and b such that

i5(3 - 2i)2 - (a - bi) = a - bi.

b. Let z1 =  √3 + i and z2 = √3 - i. Show that

z₁⁶¹ + z₂⁶¹ = 2⁶¹√3.

(Remember that: cos(- θ) = cos(θ), sin(θ) = -sin(θ) = -sin(θ), cos(2kπ + θ) = cos(θ) and sin(2kπ + θ) = sin(θ) )

<e> Evaluate the integral ∮c= 1 /(z-1)(z-2)(z-4) dz where C is |z| = 3 ?

Find the billianear transformation which maps the points z=0,1,∞ on to the points w=0,i,2i.

How do you write the residue at z=∞ where f(z) = 1/z^3+z^5?

A)-2

B)-1

C)1

D)0

Evaluate ∮c e^z/(z+2) (z+3) dz where C is |z|=1?

A)-2

B)-4

C) 1

D) 0

Give one example and also prove that for the complex series which is

1) Conditionally Convergent.

2) Absolutely Convergent.

3) Both.

4) Neither

What is the integral of 1/(z-1) (z-2) (2-4) dz where C is |z| = 3?

Give one example for the complex series which is

1) Conditionally Convergent.

2) Absolutely Convergent.

3) Both.

4) Neither.