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Integral of exponential over square root

  1. Jul 5, 2015 #1
    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. jcsd
  3. Jul 5, 2015 #2
    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.
     
  4. Jul 5, 2015 #3
    Thank you Dr.Courtney
     
  5. Jul 5, 2015 #4

    mathman

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    Science Advisor

    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.
     
  6. Jul 6, 2015 #5
    How? I'm curious :D
     
  7. Jul 6, 2015 #6
    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.##
     
  8. Jul 7, 2015 #7
    Ok thank you very much.
     
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