Consider the function f defined by f(z) =  for z ∈ ℂ such that 1 − z − z² ≠ 0. Which of the following statements is true?

Concept:

An entire function is a complex-valued function that is a complex differential in a neighborhood of each point in a domain in a complex coordinate space

Explanation:

f(z) =  for z ∈ ℂ 

singularities of f(z) is given by

1 - z - z² = 0 ⇒ z =  = 

So f(z) has a pole at z =  

So options (1) and (2) both false

f has a Taylor series expansion so

f(z) =  anzn = f(0) + z + .... 

So a0 = f(0) = 1

and a1 = 

Now, f'(z) = -(-1 - 2z)

So f'(0) = 1

So option (4) is correct and (3) false