       integral of exponential

• To: mathgroup at smc.vnet.net
• Subject: [mg104795] integral of exponential
• From: Ares Lagae <ares.lagae at sophia.inria.fr>
• Date: Tue, 10 Nov 2009 05:59:35 -0500 (EST)

```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.

Details:

* 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

```

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