Question

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

• A. $\frac{1}{1-z}$
• B. $e^{1/z}$
• C. $cos\left(z\right)$
• D. $ln\left(z\right)$

Answer 1

The correct option is C $cos\left(z\right)$

Option (a):
$\frac{1}{1-z}$, not analytic at $z=1$
Option (b):
$e^{1/z}$
$e^{1/z}=\left(\frac{1}{z}\right)+\frac{1}{2!}{\left(\frac{1}{z}\right)}^2+\frac{1}{3!}{\left(\frac{1}{z}\right)}^3+....$
not analytic at $z=0$
Option (c):
$cos\left(z\right)$
$cos\left(z\right)=1-z^2+\frac{z^4}{4!}-\frac{z^6}{6!}+\frac{z^6}{8!}$
analytic for every value of $z$, so analytic for entire $z-$ plane
Option (d):
ln $\left(z\right)$
at $z=0$, In$\left(z\right)$ is not analytical.