When to use the residue at infinity

[jack5322]

why is it that we can use the residue at infinity on the methods of contour integration example on wikipedia for the last one? we can only use the residue at infinity when it is a rational function, i.e. the ratio of two polynomials, if I'm wrong when i say this, why?

 

[jack5322]

does anyone know? please help

 

[Count Iblis]

Can you write down the example?