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$$ u_N(x,y)=\Big(\frac{8ab}{\pi^2}\Big)^2\sum^N \ \sum^N_{m,n=1,3,5,\cdots} \  \dfrac{(-1)^{\{(m+n)/2\}-1}\phi_{mn}(x,y)}{(b^2m^2+a^2n^2)}$$
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