Complex Analysis Answers

Find z1/n; for n=3, z = 1-i in C (, the Argand Plane).

If f(z) is an entire function such that f(z)=f(-z) then there exists an entire function g(z) such that f(z)=g(z^2)

Use Cauchy’s integral formula to evaluate sin(z) /((2z + 1)*3e*z) dz

1 Prove that a necessary condition that

w = f(z) = u(x, y) + iv(x, y)

be analytic in a region R is that the Cauchy-Riemann equations

∂u

∂y =

∂v

∂y ,

∂u

∂y = −

∂v

∂x

are satisfied in a region R where it is supposed that these partial derivatives are

continuous in R

Express the following in rectangular and polar form, if

Z1 = 3+ 4i

Z2= 2+3i

1. Z1*Z2

2. Z1-Z2

3. Z1/Z2

4. |Z1|

5. |Z2|

(2)if Z1=50<30°and Z2=30<60°find in rectangular

form the following

1. |Z1|

2. |Z2|

3. |Z1|-|Z2|

4. Z1*Z2

5. |Z2-Z1|

6. |Z2|/|Z1|

Show that 𝑓(𝑧) = 𝑥 ² + 𝑦 ² where 𝑧 = 𝑥 + 𝑖𝑦 is not analytic anywhere using Cauchy Riemann equations.

Develop 1/(1+z^2) in powers of z-a, a being a real number.Find the general coefficient and for a=1 reduce to simplest form.

Determine complex number(s) z such that z- i , iz- i and z-iz have the same modulus