Help getting started with this integral

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The discussion centers on evaluating the integral \(\int_{-\infty}^\infty e^{-j\sqrt{x^2+a^2}} e^{-jkx} dx\). Patrick seeks assistance in determining whether a closed form solution exists for this integral, which is crucial for expressing a polar wave as a summation of plane waves. A participant raises concerns about the convergence of the integral, suggesting that it may not have a physical meaning if it is divergent. This highlights the importance of verifying the convergence of integrals in mathematical physics.

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CuppoJava
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Hello everyone,

I'm having trouble understanding how to evaluate this integral. I am not sure whether there is a closed form solution.

[itex]\int_{-\infty}^\infty e^{-j\sqrt{x^2+a^2}} e^{-jkx} dx[/itex]

I am trying to express a polar wave as a summation of plane waves, and this integral drops out when solving for the amplitudes of the plane waves. Thanks very much for your help. A link to a reference, or the necessary techniques would be very helpful.

-Patrick
 
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Hi !

Did you first check if the integral is convergent or not ?
If not, are you sure that the given integral has a physical meaning ?
(I think that the integral is not convergent).
 
A typo in your integral ?
 

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