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Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$:

$$I(z)=\int_{S^1}|t-z|^{\beta}|dt|= \int_0^{2\pi}\bigl({1+r^2-2r\cos(\theta-\phi)}\bigr)^{\beta/2}d\theta,$$ $t=e^{i\theta}$, $z= re^{i\phi}$, $|dt| $ denote the arc length measure on $S^1$.

My question is: 1) can we, at best, evaluate this integral?

Or if not, 2) can we get $I(z)\leq K(1-|z|)^{1+\beta}$, $\beta \ne -1$, and $I(z)\leq -K\ln(1-|z|)$ if $\beta = -1$?

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    Function names like $\cos$ and $\ln$ are interpreted as strings of letters representing variables and thus get italicized if you write them like that. To get the proper font and spacing for them, you can use the predefined commands like `\cos` and `\ln` to get $\cos$ and $\ln$, respectively, or, if you need a function for which there's no predefined command, `\operatorname{name}` to get $\operatorname{name}$.2012-10-21
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    Note that you can get rid of $\phi$ by transforming to $\psi=\theta-\phi$.2012-10-21
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    I found the bad typesetting so distracting that my brain refused to consider the mathematical content. Fixed.2012-10-21
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    The estimate is certainly wrong when $\beta>0$, since the integral approaches a positive limit when $r\to1$ in that case. From your final question, I gather that you are primarily interested in the case $\beta<0$; but it might be worth a mention anyhow.2012-10-21

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