A small problem with a complex integral

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SUMMARY

The discussion centers on evaluating a complex integral over a semicircular contour in the upper half-plane, specifically the integral of \(\frac{(\ln x)^2}{1+x^2}\). The integral is expected to yield a result of zero. Participants highlight the importance of considering the arc-length of the contour, which is \(\pi r\), as the radius approaches zero, impacting the evaluation of the integral.

PREREQUISITES
  • Understanding of complex analysis, specifically contour integration.
  • Familiarity with logarithmic functions and their properties.
  • Knowledge of limits and behavior of functions as variables approach zero.
  • Basic grasp of arc-length calculations in the context of integrals.
NEXT STEPS
  • Study the principles of contour integration in complex analysis.
  • Learn about the properties of logarithmic functions in integrals.
  • Explore the concept of limits in the context of complex functions.
  • Investigate the relationship between arc-length and integral evaluation.
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in advanced integral calculus will benefit from this discussion.

krishna mohan
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Hi...

I have an integral over a contour. The contour is a semicircle with vanishing radius around the origin and situated in the upper half plane.

The integrand is [tex]\frac{(lnx)^2}{1+x^2}[/tex].

The integral is supposed be zero.


I don't see how. Taking the modulus and letting the radius go to zero, I find that the denominator goes to 1 and the numerator is of the form [tex]ln(r^2)+\theta^2[/tex] with r going to zero.

Can anybody tell me what I might be missing?
 
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hi krishna! :smile:

but the arc-length is πr, so you multiply by that :wink:
 
Yep..thanks a lot tiny tim!:smile:
 

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