Residue Theorem: Theory & Application

[mmzaj]

the Residue theorem states that :
$$\oint {f(z)dz}$$ = 2$$\pi i$$$$\sum Res f(z)$$and the summation is taken for all the poles of f(z) enclosed by the counter at which the integration is performed .

now i have read somewhere that

$$\oint \frac{f(z)dz}{z^{n+1}}$$ = 2$$\pi i$$[tex]\sum Res f(z) $$a^{n}$$

 

[Pere Callahan]

What's the question? And you messed up the second math disply.

 

[mmzaj]

i'm sorry ! it took me half an hour writing up , and i don't know how it got posted , but it really looks bad :) .
anyway ... my question is : for the second integral - the one with z raised to n+1 in the denominator - is it possible to evaluate it using the Residue theorem ? what i have read that it can be evaluated using a series in which each pole is raised to n and multiplied with it's residue .
again , I'm very sorry , but latex needs to improved deeply .

 

[mmzaj]

come on guys ... !

 

[mmzaj]

ok , now i got things going right . for a function f

$$\oint f(z)dz$$ = 2$$\pi i$$ $$\sum Res(f,z_k)$$

if f is a rational function , does the following relation hold ??

$$\oint z^n f(z)dz$$ = 2$$\pi i$$ $$\sum Res(f,z_k)$$ $$\ {z_k}^n$$

where $$\ z_k$$ are the poles of f .

any help is appreciated .