homework assignment in "complex functions"

1. Calculate the following:

2. Find all the solutions for the following equations:

3. Given that:

Prove that:

4. Assuming the radius of convergence of the power series ∑ 𝑐 𝑧 is R. Calculate the radius of convergence for the following power series:

5. Calculate the radius of convergence for the following power series:

6. Find the Holomorphic function f(z) if the following is given:

a . b . c . d . e . f .

7. Prove that if 𝑓(𝑧) and 𝑓̅(𝑧) are Holomorphic functions in domain D, than 𝑓(𝑧) is a constant function.

8. Prove that if 𝑓(𝑧) and 𝑔(𝑧) are Holomorphic functions in domain D, than

𝐼𝑚 𝑓(𝑧) + �̅�(𝑧) = 0 if and only if 𝑓(𝑧) − 𝑔(𝑧) = 𝐶 in domain D and C is a Real constant.