Complicated integration of complex number

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SUMMARY

The forum discussion focuses on the integration of complex functions using the Cauchy Integral Formula (CIF) and the Residue Theorem. Participants clarify that CIF can be applied even when poles exist inside the contour, provided the function is analytic elsewhere. The discussion emphasizes the importance of identifying singularities and using appropriate derivatives when applying these theorems. The residue theorem is highlighted as a comprehensive method that encompasses both CIF and Cauchy's Integral Theorem (CIT).

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  • Familiarity with analytic functions and singularities in complex analysis.
  • Knowledge of differentiation techniques for complex functions.
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  • Study the application of the Residue Theorem in complex integration problems.
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  • #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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