Why Does lnZ Not Represent an Isolated Singularity?

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Homework Help Overview

The discussion revolves around the concept of isolated singularities in complex analysis, particularly focusing on the logarithm function and its branches. Additionally, there is a mention of an integration problem involving logarithmic functions and branch cuts.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore why the logarithm function cannot be holomorphic in certain domains due to its multiple branches. There are also inquiries about defining branch cuts in the context of a specific integral involving logarithmic terms.

Discussion Status

The discussion includes attempts to clarify the nature of logarithmic functions and their singularities. Some participants are seeking guidance on how to approach the integration problem and the concept of branch cuts, while others have pointed out the importance of identifying poles in the integrand.

Contextual Notes

There is a repeated emphasis on the need for participants to post their attempts and questions in a structured manner, as well as a suggestion to start new threads for separate inquiries. The context of homework constraints is evident in the nature of the questions being asked.

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why lnZ is not isolated singularity?
 
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Because you can't define ln(z) to be holomorphic in the disk after removing 0. It has multiple branches.
 
Questions on Poles

I am trying to do my homework on:

Integration form 0 to infinity of [( log x)^4 ]/[1+x^2] dx =

by first defining what is a branch cut and how it will fit into the above integrand.

Thanks

Marie
 


logmarie said:
I am trying to do my homework on:

Integration form 0 to infinity of [( log x)^4 ]/[1+x^2] dx =

by first defining what is a branch cut and how it will fit into the above integrand.

Thanks

Marie

I don't recommend posting a brand new question onto an existing thread. Start a new one. While you're doing that figure out how to start the problem so you can post an attempt. Where are the poles?
 
How do I start the new threads?

Thanks
 
The poles is x= +/- i
 
logmarie said:
How do I start the new threads?

Thanks

Go to the Calculus and Beyond forum and under Forum Tools you'll find start new thread.
 

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