According to my lecture notes, $z^*$ has an essential singularity (asterix denotes complex conjugate). However, it is not explained why nor at which point. Can anyone elaborate where the singularity is?
The reason why I ask is because $$ \oint(z^* dz) \neq 0 $$
where the curve is a circle centered at the origin, which must mean that indeed there is a singularity.