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A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is:

$$\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$$ Taken separately the integrals have solutions:

$$\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}~\right)}\mathrm{d}z = b~ \mathrm{K}_1 ( \lambda b , \textrm{sinh}^{-1}(a/b) ) $$ where the $K_1$ is an incomplete modified Bessel function of the second kind.

Can anyone think of a way to extend this in the case where there are two square roots in the exponential?


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I have a paper (unpublished) precisely on this topic. Drop me a line ( to obtain a copy. – larry Aug 10 '13 at 20:40
What is incomplete modified Bessel function of the second kind? – Harry Peter Oct 23 '13 at 12:53

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