Poisson kernel

1. Jul 14, 2012

hedipaldi

1. The problem statement, all variables and given/known data

show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a)

2. Relevant equations

the integral of poisson kernel from -pi to pi is 2pi

3
. The attempt at a solution
:it must by a direct conclusion of the integral in 2 but somehow i fail to see it.

2. Nov 2, 2012

matematikuvol

Is $r$ always positive? Is $P(-r,x)$ defined?

3. Nov 2, 2012

hedipaldi

main guidelines for the proof in the attachment.

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4. Nov 2, 2012

hedipaldi

I am hedipaldi but i did not pause this question.How come it is on my name??