Consider the function f defined by f(z) = \(\rm\frac{1}{1−z−z^2}\) 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) = \(\rm\frac{1}{1−z−z^2}\) for z ∈ ℂ 

singularities of f(z) is given by

1 - z - z² = 0 ⇒ z = \({-1 \pm \sqrt{1+4} \over 2}\) = \({-1 \pm \sqrt{5} \over 2}\)

So f(z) has a pole at z = \({-1 \pm \sqrt{5} \over 2}\) 

So options (1) and (2) both false

f has a Taylor series expansion so

f(z) = \(\rm\displaystyle\sum_{n=0}^{\infty}\) anzn = f(0) + \(\frac{f'(0)}{1!}\)z + .... 

So a0 = f(0) = 1

and a1 = \(\frac{f'(0)}{1!}\)

Now, f'(z) = -\(\rm\frac{1}{(1−z−z^2)^2}\)(-1 - 2z)

So f'(0) = 1

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