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Hadamard Integration

Dear All,

The following is about a two dimensional integral that appeared on a
private communication with a member of the forum.

Known constants:
params = {k -> 0.1, a -> 1, M -> 0.3, Deltax1 -> 1};
e1 = 1;
b = Sqrt[1 - M^2];
x0 = a/2;

h is the variable of integration of the external integral (the regular

y0 = -h;
z0 = 0;
r = Simplify[Sqrt[y0^2 + z0^2], h > 0]
R = Sqrt[x0^2 + b^2*r^2]
X = (x0 - M*R)/b^2
X1 = Sqrt[X^2 + r^2]

The Hadamard integral is the internal integral with variable of
integration u.

ff = (1/(8*Pi))*(Deltax1*NIntegrate[((M*E^(I*k*X))/(R*X1) +
NIntegrate[E^(I*k*u)/(r^2 + u^2)^(3/2), {u, -Infinity, Plus[X]}])/
E^(I*k*x0), {h, -e1, Plus[e1]}])

Some work has already be done on ii but I really appreciate any
Best Regards

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