C-R of Laurent Series

[hob]

When doing complex contour integration one can use the C-R formula or the Laurent series and find the first coefficient of the principle part. What are the selection criteria for choosing these methods?

Regards,

Hob

 

[HallsofIvy]

What do you mean by "use the Cauchy-Riemann equations to do a contour integral"? Are you referring to actually integrating around the contour as opposed to summing the residues inside the contour?

 

[hob]

Hi, I mean summing within the contour.

$$2 \pi$$ x $$\sum residues$$ = Integral around the contour.

You can also use the Laurent Expansion and finding the first principle coefficient.

I am unsure what method you would use when presented with a complex integration,

Regards,

 

[HallsofIvy]

??The "first principle coefficient" of the Laurent expansion, around pole $z_0$, by which I think you mean the coefficient of $z^{-1}$, is the residue at that $z_0$. They are the same method.

 

[zhentil]

Correct me if I'm wrong, but I think you mean should you try to evaluate the integral directly, or do the residue by other means? Almost certainly the residue method. After all, if you could muscle your way through integrals, you wouldn't need contour integration.