# How to calculate residue of a function dependent on conjugate argument?

How to calculate residue of a function that is dependent on $\bar{z}$ at singularity $z_0$? $$\operatorname{Res}_{z_0} f(\bar{z})$$

How to calculate residue of a function that is dependent both on $\bar{z}$ and on $z$ at singularity $z_0$? $$\operatorname{Res}_{z_0} f(z,\bar{z})$$

Say, $$f(z)=\frac{1}{\sin (\bar{z})+\frac{1}{2}}$$ or $$f(z)=\frac{1}{\bar{z} \left(\sin (z)+\frac{1}{2}\right)}$$

I can calculate residues if there is only $z$ but cannot figure out, if it is possilbe also with $\bar{z}$, not sure whether it has a sense in the first place.

There's no such thing. A function $f$ with an isolated singularity at $p$ has a residue at $p$. Saying $f$ has an isolated singularity at $p$ means it's holomorphic in some punctured disk $0<|z-p|<r$. So the functions you name do not have isolated singularities, hence asking about their residues simply makes no sense.
• So for example $z_0=-\frac{\pi }{6}$ is not an isolated singularity for the two defined functions? I though it is... – azerbajdzan Nov 12 '17 at 15:19