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Re: A definite integral

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On Tue, 7 Feb 2023 07:32:01 -0800 (PST), in article <ac6dded8-1440-4858-9ef8-
3fc8bb634f15n@googlegroups.com>, Robert Gragg wrote:
> 
> Can Maple provide a result for the integral
> \int_{-\pi/2}^{\pi/2} e^{-i a cos\phi} \cos^2\phi d\phi
>  with a>0?
> Or maybe an asymptotic result for  a>>1?


Per your question, I gave Maple this:

  f := a -> int(exp(-I*a*cos(phi))*cos(phi)^2, phi = -1/2*Pi .. 1/2*Pi)

Maple claimed that

  f(a) = Pi*(-a*StruveH(0, a)*I + a*BesselJ(0, a) + 
         StruveH(1, a)*I - BesselJ(1, a))/a

and futhermore that

  limit(f(a), a = infinity) = 0

The last equality is heuristically supported by:

  seq(abs(evalf(f(10^n))), n = 1 .. 5) = 
    0.7917525236, 
    0.2506651220, 
    0.07926654025, 
    0.02506628274, 
    0.007926654597

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A definite integral Robert Gragg <robertfgragg@gmail.com> - 2023-02-07 07:32 -0800
  Re: A definite integral William Unruh <unruh@invalid.ca> - 2023-02-08 06:53 +0000
  Re: A definite integral Wasell <wasell@example.com> - 2023-02-09 17:32 +0100

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