Nasty Integral

This one has me baffled:

∫e^{-√(x2 + a2)}dx

I made the obvious substitution:

y = √(x² + a²)

and came up with:

∫dy y e^-y/√(y² - a²)

Integrating by parts results in:

e^-y√(y² - a²) + ∫e^-y dy √(y² - a²)

and got stuck. Any ideas?

Quote:

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Originally Posted by VBAhack

This one has me baffled:

∫e^{-√(x2 + a2)}dx

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I was gonna say it can't be solved analytically but I first tried with Mathematica, just in case, and it could not solve it.

krtxmrtz,

Thanks for the try. The more I stared at it the more I thought the same thing - there's just no way to get rid of the pesky exponential. Anyway, I've since been told that this is a Bessel function of the 2nd kind, although I haven't taken the time to try to verify that.

Quote:

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Originally Posted by VBAhack

krtxmrtz,

Thanks for the try. The more I stared at it the more I thought the same thing - there's just no way to get rid of the pesky exponential. Anyway, I've since been told that this is a Bessel function of the 2nd kind, although I haven't taken the time to try to verify that.

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I've never been into Bessel functions that much but I think they must be calculated by numerical methods.
If you have Mathematica or similar software it's good to first have it give a try on the integral. If it can't be solved you'll spare your head quite a few rams into a concrete wall. ;)

Thanks for the good advice. The concrete wall got rather dented with this one....... :)