Which one of the following functions is analytic over the entire complex plane?

Question

Question

This question was previously asked in

GATE EC 2019 Official Paper

Answer 1

(i) ln z

At, z = 0

The function is not analytic

(ii) e^1/z

Using the expansion formula for e^x

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots$

$e^{1 / z} = 1 + (\frac{1}{z}) + \frac{1}{2!} (\frac{1}{z^{2}}) + \frac{1}{3!} (\frac{1}{z^{3}}) + \dots$

At, z = 0, e^1/z is not analytic

(iii) $\frac{1}{1 - z}$

At, z = 1, the function is not analytic

(iv) cos z

expansion of cos z is:

$\cos z = 1 - \frac{z^{2}}{2!} + \frac{z^{4}}{4!} - \frac{z^{6}}{6!} + \dots$

The function cos z is analytic over the entire z.