Poisson Integral Formula

Assume that u(r,Q) = R(r)0(0). Substituting into the equation and rearranging, we get 

To ensure that the solution is finite at r = 0, D = 0. Thus, by the superposition principle, the general solution is given by 

The solution of the Laplace equation in a unit sphere with prescribed on the surface of the sphere is given by 

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The semiclassical approximation is reached in three different steps (Berry and Mount, 1972). The first stage is to transform the summation over the discrete values of / into an integration over the continuous variable A = (/ + ). The proper way to do this is to use the Poisson summation formula which gives... 

In the practical matter of performing the summations indicated for the various formulas that must be evaluated, the question arises as to how many terms need to be included this question is analogous to the need to decide the limits of integration that was implicit in evaluating the analogous expressions for the Normal Distribution. In the case of the Poisson distribution this is one decision that is actually easier to make. The reason is... 

Firstly, the Poisson formula (4.26) is employed to provide the extended useful integral formulation... 

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