Residue of ArcTan: Is the Residue Theorem Applicable?

[Pere Callahan]

Hi Folks,

Does it make sense to speak of the residue of the arctan function at $$z=\pm i$$?

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...

 

[Office_Shredder]

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

 

[tacman]

You are correct in your assumption that the Residue Theorem cannot be applied to functions that are not holomorphic in a small disk around the singularity. In order for the Residue Theorem to be applicable, the function must be holomorphic in the region of the contour and have isolated singularities within that region. Since the arctan function and the natural logarithm are not holomorphic in a small disk around their singularities, the Residue Theorem cannot be used in these cases.

However, it is still possible to calculate the residues of these functions using other methods, such as the Laurent series expansion. It is always a good idea to refer back to your Complex Analysis textbooks for a deeper understanding of these concepts.