Complicated integration of complex number

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Homework Help Overview

The discussion revolves around a complex integration problem involving the Cauchy integral formula and residue theorem. Participants are exploring the evaluation of integrals with isolated singularities and the implications of poles within a contour.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the use of the Cauchy integral formula and the residue theorem, questioning the necessity of decomposing fractions and the implications of poles inside the contour. There are attempts to clarify the nature of singularities and how they affect the choice of methods for integration.

Discussion Status

Some participants have provided guidance on identifying functions that are analytic within the contour and applying the Cauchy integral formula. There is an ongoing exploration of different approaches, including the residue theorem, and participants are seeking clarification on their understanding of the concepts involved.

Contextual Notes

Participants are navigating the complexities of poles and singularities, with some expressing uncertainty about the correct application of theorems and formulas in the context of their specific problem. There is a mention of homework constraints and the need for careful consideration of the functions involved.

  • #31
CAF123 said:
Yes, but you already done the computation using CIF so need to bother doing it again.

The it means the this wasy is no simpler than CIF? Because I still haven't finished the derivation.
 
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  • #32
MissP.25_5 said:
The it means the this wasy is no simpler than CIF? Because I still haven't finished the derivation.
Well, it is a matter of taste but in some sense I find the residue theorem easier to actually get to that equation involving the derivative. You did not have to, for example, study f(z) to find a part that is analytic on the interior of the contour. Even though that is still quite a quick step...

Laurent expansions of functions also give you the residue - it is defined as the coefficient of the 1/z term (the first term in the expansion that 'knows' about the singularity). But again, you end up with the final equation and in fact this is how the general formula for the residue of a nth order pole was derived. If you have a textbook or notes, check it out.
 

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