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Integral of exponential

  1. Nov 9, 2009 #1
    Hello all,

    I am trying to solve an integral with Mathematica, but I do not succeed. I am wondering whether the integral cannot be solved, or whether Mathematica cannot solve the integral, or whether I am doing something wrong.


    * Mathematica does not seem to be able to solve the integral below:

    With[{fx = fr Cos[ft],
    fy = fr Sin[ft]}, \[ExponentialE]^(-2 \[Pi]^2 ((f^2 + fx^2) sx^2 +
    fy^2 sy^2)) Cosh[4 f fx \[Pi]^2 sx^2]]
    Integrate[%^2, {ft, 0, 2 Pi}]

    * For sx = sy, Mathematica can solve the integral:

    With[{fx = fr Cos[ft], fy = fr Sin[ft],
    sx = sy}, \[ExponentialE]^(-2 \[Pi]^2 ((f^2 + fx^2) sx^2 +
    fy^2 sy^2)) Cosh[4 f fx \[Pi]^2 sx^2]]
    Integrate[%^2, {ft, 0, 2 Pi}]

    \[ExponentialE]^(-4 (f^2 + fr^2) \[Pi]^2 sy^2) \[Pi] (1 +
    BesselI[0, 8 f fr \[Pi]^2 sy^2])

    * I am suspicious because Mathematica also does not solve the following known integral:

    Integrate[Exp[x Cos[t] + y Sin[t]], {t, 0, 2 Pi}]

    which equals

    2 \[Pi] BesselI[0, Sqrt[x^2 + y^2]]

    * However, Mathematica does solve the integral below

    Integrate[Exp[x Cos[t]], {t, 0, 2 Pi}]

    2 \[Pi] BesselI[0, x]

    Any help and/or insight is appreciated.

    Best regards,

    Ares Lagae
  2. jcsd
  3. Nov 9, 2009 #2


    User Avatar
    Science Advisor

    latex doesn't seem to be working for you.
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