complex integration

[Angela1]

hey guys,

i just have a question regarding this question,

what i have done is just used cauchys integral formula
so for n = 0, its just -2iPi right? because its 1/(5-z)=- 1/(z-y) and c is a close simple curve

then for n=1,4 just let f(w) = 1/z or 1/z^4

so then its 2iPi f(w)= 2iPi 1/5, 2iPi 1/5^4,

It seemed to straight forward to me so im paranoid about the answer,

for n=n would it juts be

2iPi 1/5^n ?

thanks alot in advance

 

[galactus]

Yes, you are correct. Very good.

You can also see it by looking at the coefficient of the 1/z term in the Laurent expansion.

i.e. if n=2, then we have \(\displaystyle \frac{1}{z^{2}(5-z)}=\frac{1}{5z^{2}}+\frac{1}{25z}+....\)

See?. The coefficient of the 1/z term is \(\displaystyle \frac{1}{5^{2}}\)

And so on.