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Integrate problem with undefined constant

  1. Jul 29, 2011 #1
    Hello!

    I'm trying to calculate this integral:

    Integrate [Exp[-I a *y] Exp[I*Sin[y]] Exp[-y*Sign[y]], {y, -Infinity, Infinity}] , where "a" - constant that i'll define later.

    So the result of the calculation acсording to Mathematica is the same expression : Exp[-I a *y] Exp[I*Sin[y]] Exp[-y*Sign[y]].

    What can i do to calculate my integral without defining my constant "a" and using NIntagrate?

    Thank you in advance.
     
  2. jcsd
  3. Jul 29, 2011 #2
    I doubt that Mma returns the integrand - it probably returns an unevaluated integral, no matter what choice you make for the constant a. The integral probably does not have a nice closed form in terms of any common or existing special functions.

    Mma doesn't even evaluate the "simpler" integral Exp[I*Sin[y] - y^2].

    If you just need to use this result just define the numerical integral (with memoization)

    int[a_?NumericQ] := int[a] = NIntegrate[Exp[-I a*y] Exp[I*Sin[y]] Exp[-y*Sign[y]], {y, -Infinity, Infinity}]//Chop

    It's a little slow... but you can get a reasonable plot of the integral

    Plot[int[x], {x, -10, 10}, PlotPoints -> 15, MaxRecursion -> 3, PlotRange -> All]
    attachment.php?attachmentid=37617&stc=1&d=1311986079.png
     

    Attached Files:

  4. Jul 30, 2011 #3
    Thank you so much for answer!
     
  5. Jul 31, 2011 #4
    Not a problem. I hope it was useful.
    If you do find a closed form to that integral - don't forget to post it here!
     
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