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Hi,
thanks to Raymond for his observation, that
u(a,b)=int_0^oo exp( I * b * x - a * sqrt( 1 + x^2)) dx
should solve the Helmholtz equation.
The general solution to (d_aa + d_bb - 1) u(a,b) =0 is
u(a,b) = (A*besselJ(m,-I*sqrt(a^2+b^2)+B*besselY(-I*sqrt(a^2+b^2)))*
(exp(+/- I * m * arctan(b/a))), obtained by separation into
polar coordinates and backsubstitution.
For m=0, the 2nd factor is (1+C*arctan(b/a)).
If I set A=pi/2 * (-I)^(m+1) and B=-pi/2 * (-I)^m the first factor becomes
(besselK(m,sqrt(a^2+b^2)). Then the cosine part of the integral is readily
identified with m=1 :
int_0^oo cos(b*x)*exp( - a * sqrt( 1 + x^2)) dx =
a/sqrt(a^2+b^2)*besselK(1,sqrt(a^2+b^2));
But I still cannot find the sine part
int_0^oo sin(b*x)*exp( - a * sqrt( 1 + x^2)) dx, which falls off with
increasing b significantly weaker than the besselk(m,sqrt(a^2+b^2)).
Maybe there is another linear combination of bessel functions to be used ?
Clueless,
Andreas
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