Residue of ArcTan: Is the Residue Theorem Applicable?

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SUMMARY

The discussion centers on the applicability of the Residue Theorem to the arctan function at z=±i and the natural logarithm at z=0. Participants conclude that the Residue Theorem cannot be applied to these functions due to their non-holomorphic nature in any neighborhood around their singularities. Specifically, the arctan function and the logarithm require careful consideration of their domains, as the logarithm necessitates a branch cut that disrupts holomorphicity around z=0. Thus, the Residue Theorem is inapplicable in these cases.

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Pere Callahan
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Hi Folks,

Does it make sense to speak of the residue of the arctan function at [tex]z=\pm i[/tex]?

Or the residue of the natural logarithm at z=0 ..?

The problem probably is that these functions are not holomorphic in however a small disk around the singularity...

So am I right in assuming that the Residue Theorem cannot be applied to such functions?

Maybe I should read up on this in my old Complex Analysis textbooks...:rolleyes:
 
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I don't know about arctan off the top of my head, but about log, it doesn't. The reason being that Laurent series requires you to have an annulus around the point in which the function is holomorphic. To define log, you have to take a cut plane, so any annulus around 0 has the cut running through it, and hence log isn't holomorphic at that point
 

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