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In Equations 7 and 9 the potential is expressed using complex notation for convenience. However, the result is necessarily real because the contributions from the m,n and -m-n terms in the summation are so that the imaginary part is zero.
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In Equations 7 and 9 the potential is expressed using complex notation for convenience. However, the result is necessarily real because the contributions from the m,n and -m-n terms in the summation are so that the imaginary part is zero.
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2d is taken to zero when d tends to zero. This is not a valid limit for an arbitrary m,n pair as these Fourier indices can be arbitrarily large. However, because of the exponential decay in z, one can always find an m,n pair of values such that all potential terms above it are arbitrarily small. Therefore, the limit needs to be taken in practice only for finite values of m,n and is therefore correct.
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