Integral of exponential over square root

1. Jul 5, 2015

newgate

Hello guys,
How can I evaluate the following integral please?
$\int_{-\infty}^{+\infty}\frac{e^{i A x}}{\sqrt{x^2+a^2}}dx$
Thank you

2. Jul 5, 2015

Dr. Courtney

Use Euler to expand the complex exponential into cos() + i*sin().

Separate into two separate integrals. The integrand with sin() is odd, so its integral is ZERO.

Wolfram Alpha won't work the cos() integrand with a and A as symbols, but with a bit of plugging in numbers, one can quickly determine that the integral of the cos() integrand is 2*K0(a*A), where K0 is the modified Bessel function of the second kind. A good integral table would likely tell you that also.

Mathematica or Wolfram Alpha Pro would probably do the original integral.

3. Jul 5, 2015

newgate

Thank you Dr.Courtney

4. Jul 5, 2015

mathman

I am not familiar with Wolfram. However, it is easy enough to modify the integral to get rid of A or a, so that you have only one constant.

5. Jul 6, 2015

newgate

How? I'm curious :D

6. Jul 6, 2015

micromass

Substitution $u = Ax$. Then you just have to find an integral of the type $\int_{-\infty}^{+\infty} \frac{e^{ix}}{\sqrt{x^2 + C}}dx.$. Likewise, a good substitution will leave you with an integral of the type $\int_{-\infty}^{+\infty} \frac{e^{Cix}}{\sqrt{x^2 + 1}}dx.$

7. Jul 7, 2015

newgate

Ok thank you very much.

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