Suppose $f$ has an essential Singularity at $z = a$ and $g$ has a pole at $z = a$. Then the product $fg$ has an essential Singularity at $z = a $. Is this hold if $g$ has removable Singularity $ z= a$ or $g$ is analytic at $z = a$
Since $f$ has an essential Singulalty at $z = a$,then $f$ can be expressed as a laurent Series. i.e $$f(z) = \sum_{i= -\infty}^{\infty} a_i (z-a)^i$$
Suppose $g$ has a ploe of order m, then $g$ can be written as
$$g(z) = \sum_{j = -m }^{\infty} b_j (z-a)^j$$.
Please tell me how to proceed further. Thank you.
$e^{1/z}$ has an essential singularity at $z=0$. If you multiply it by a constant function (entire/analytic), or a removable discontinuity, it will remain an essential singularity. You would need to multiply it by something quite extraordinary to be able to get rid of the essential singularity.