A function f: ℂ → ℂ is said to be analytic at ∞, if the function g defined by

g(w) = f ($$\frac{1}{w}$$​) is analytic at 0 with an appropriate value given for g(0).

Which of the following statements is true?

Correct option is C

Option (C):

 $$f(z)=e^{\frac{1}{z-z_0}}, \ f.s\ z_0\in \mathbb{C}\\[10pt]\text{Checking analyticity at } \infty \\[10pt]g(w)=f(\frac{1}{w})=e^{ \frac{1}{\frac{1}{w}-z_0}} \\[10pt]\implies g(w)=e^{\frac{w}{1-wz_0}}\\[10pt]$$

We know that exponent is an entire function.

So, g(0) gives an appropriate value and g is analytic at 0.

$$\implies$$​ f is analytic at $$\infty$$

$$\implies \textbf{Option C is correct}$$​​