What Is the Deeper Interpretation of a Complex Residue?

[skook]

Just spent the last few months working on an undergrad course in complex analysis and have a couple of things that aren't clear to me yet. One of them is the meanings of the residue of a complex function. I understand how to find it from the Laurent series and using a couple of other rules and I understand how it works with the residue theorem. But I still feel like there is a deeper interpretation out there waiting for me...perhaps it's something geometrical?

Grateful for any enlightened comment.
:-)

 

[lurflurf]

There are several equivalent ways to think about it. One is as the 1/(z-a) term of the laurent expansion. Another is the inner product of f with 1/(z-a). One could also think of it as the amount of (order 1) infinity at the point a.

 

[skook]

Thanks for that...the nearest I can get is that it could be 1/(2*pi) of a Dirac delta function with a pi/2 twist.
:-)