- #1

curious_mind

- 20

- 7

- Homework Statement:
- Calculate the residue of the complex function ## f(z) = \dfrac{\ln z}{z^2 + 4} ## at ## z = 2 e^{i \pi} ##

- Relevant Equations:
- ## a = \lim_{z \rightarrow z_0} \left[ \left(z_0 - z\right) f(z) \right] ##

This question is given as an example in the book by

Arfken, Weber, Harris, Mathematical Methods - a Comprehensive Guide, Seventh Edition.

It is solved as below attached in the image.

Can someone point it out how they proceed with calculations ? I do not seem to get their calculation.

I am aware ## \ln z ## is a multivalued function. But at this point I do not know things about Branch points and etc.

According to my understanding the function is not singular at point ## z=2 e^{i \pi} =−2 ## . So why they have used limits ?

Am I missing something ? Please help.

Thanks.

Arfken, Weber, Harris, Mathematical Methods - a Comprehensive Guide, Seventh Edition.

It is solved as below attached in the image.

Can someone point it out how they proceed with calculations ? I do not seem to get their calculation.

I am aware ## \ln z ## is a multivalued function. But at this point I do not know things about Branch points and etc.

According to my understanding the function is not singular at point ## z=2 e^{i \pi} =−2 ## . So why they have used limits ?

Am I missing something ? Please help.

Thanks.