1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ So for example $z_0=-\frac{\pi }{6}$ is not an isolated singularity for the two defined functions? I though it is... $\endgroup$ – azerbajdzan Nov 12 '17 at 15:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.