with an as yet unspecified split point at distance 0 < [tau] < 1 from the center of the Gaussian in the integrand
We only need to expand again the integrand
of (30) (with some care) and collect the main terms:
Near such a point the exponential factor in the integrand
This is because the integrand
in (5) is undefined at [r.
0] - [epsilon] < 2 , the integrand
is of order [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] and we can safely shift the line of integration to [C.
The proof of Proposition 2 consists of three steps: (i) the Poisson expectation is first expressed in integral form (Equation (9)); (ii) the integrand
is next analysed by means of the Mellin transform (Lemma 1); (iii) the outcome of the local Mellin analysis is finally used to estimate the Poisson expectation.
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This estimate involves second order derivatives of the integrand
function f(t) = [([t.
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We have to face the problem that, both, the series and the integral are not absolutely convergent since the integrand
is only of order 1/s.
If there is more than one root, then the contribution from each root should be included; if none, then that term of the integrand
is replaced by zero.
The derivation is accomplished by taking the dot product of the integrand
of the stress equilibrium equation (Eq 10) with a virtual displacement vector and then integrating the resulting expression over the volume.