Lemma 3.1 [031U]

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Lemma 3.1

Let

T_{j}={1\over 4\pi}\sum_{n=-j}^{j}\left({1\over\sqrt{(u+n\epsilon)^{2}+y_{1}^{2}+y_{2}^{2}}}-a_{|n|}\right)

where

a_{n}=\cases{1/n\epsilon&$n\not=0$\cr 2(-\gamma+log(2\epsilon))/\epsilon&$n=0$\cr}

and $\gamma$ is Euler’s constant. Then

(a) the sequence $\{T_{j}\}$ converges uniformly on compact sets in $D\times{\bf R}-\{0\}\times\epsilon{\bf Z}$ to a harmonic function $V_{0}$ . Here $D\subseteq{\bf C}$ is the unit disc centred at the origin.

(b) $V_{0}$ has an expansion, valid when $|y|\not=0$ ,

V_{0}=-{1\over 4\pi\epsilon}\log|y|^{2}+\sum_{m=-\infty\atop m\not=0}^{m=\infty}{1\over 2\pi\epsilon}e^{2\pi imu/\epsilon}K_{0}(2\pi|my|/\epsilon)

where $y=y_{1}+iy_{2}$ and $K_{0}$ is the modified Bessel function. (See [3], pg. 374.)

(c) There exists a constant $C$ such that for any $0<r_{0}<1$ , there exists an $\epsilon_{0}>0$ such that for all $\epsilon<\epsilon_{0}$ , $|y|>r_{0}$ ,

\left|V_{0}+{1\over 4\pi\epsilon}\log|y|^{2}\right|\leq{C\over\epsilon}e^{-2\pi|y|/\epsilon}.

(d) If $r\leq 1$ , and $f$ is a harmonic function on the disc $D_{r}$ of radius $r$ such that $f(y)-{1\over 4\pi}\log|y|^{2}>0$ for $|y|\leq r$ , then there exists an $\epsilon_{0}$ such that for all $\epsilon<\epsilon_{0}$ ,

V_{0}+f(y)/\epsilon>0

in $D_{r}\times{\bf R}$ .