decomposition of a logarithm of 1/R in the series

eugene · science forum Guru Joined: 24 Nov 2005 Posts: 331

I'ver encountered this problem while calculating the logarithmic potential

V(M) = V(r) = int_{circle of radius a} log(1/R) dP, where

r is the distance between the given point M and the point in the circle P, R = MP, in other words thi integral maybe written as

V(r) = int_[0,2pi] int_[0,a] log[ 1/sqrt(r^2 + l^2 -2lr*cos(alpha) ) ]*l dld(alpha).

And now i saw a formula of decomposition of log(1/R) as

log(1/R) = log(1/r) + sum_{n=1^infty} 1/n (l/r)^n cos(nalpha) .

Could you please explain how they got this formula ?
How can i prove it.

Thanks

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