cauchy residue theorem question

use the Cauchy residue theorem show that for any integer ,

where is a circle of radius unity centred at

this part was easy enough.

(by Cauchy residue theorem)

if not 0, or if

now i have to use this result to deduce that, if you are given

then

and

where is the derivative of with respect to z

i have no idea how to even start the second part of this question!! any tips??

Re: cauchy residue theorem question

Quote:

Originally Posted by

**wik_chick88** use the Cauchy residue theorem show that for any integer

,

where

is a circle of radius unity centred at

this part was easy enough.

(by Cauchy residue theorem)

if

not 0, or

if

now i have to use this result to deduce that, if you are given

then

and

where

is the derivative of

with respect to z

i have no idea how to even start the second part of this question!! any tips??

I would say, fix an and note then that . Thus, recalling that power series are uniformly convergent, so you can exchange integral and series you get

Re: cauchy residue theorem question

thankyou so much! with your help it was easy to do the second bit as well the part so thankyou thankyou thankyou!

Re: cauchy residue theorem question

this is a different question but is the same kind of stuff...

you are given

where is a circle of radius unity centred at and is a circle of radius unity centred at and

with being a constant and being the derivative of with respect to

show that

any ideas????