       Re: troublesome integral

• To: mathgroup at smc.vnet.net
• Subject: [mg126011] Re: troublesome integral
• From: "Stephen Luttrell" <steve at _removemefirst_stephenluttrell.com>
• Date: Fri, 13 Apr 2012 04:44:42 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jm64ib\$6la\$1@smc.vnet.net>

```You can rewrite the integral as

Integrate[
Cos[\[Alpha] + \[Beta]] Exp[I z Cos[\[Beta]]], {\[Beta], 0, 2 \[Pi]},
Assumptions -> z \[Element] Reals]

which can be rearranged as (the Sin[\[Alpha]] Sin[\[Beta]] part integrates
to zero)

Integrate[
Cos[\[Alpha]] Cos[\[Beta]] Exp[I z Cos[\[Beta]]], {\[Beta], 0,
2 \[Pi]}, Assumptions -> z \[Element] Reals]

which evaluates to

2 I \[Pi] BesselJ[1, z] Cos[\[Alpha]]

Over the years I too have encountered this problem with evaluating integral
representations of Bessel functions, and the only solution I have found is
to give mathematica a helping hand, as above.

--
Stephen Luttrell
West Malvern, UK

"peter lindsay" <pl0 at me.com> wrote in message
news:jm64ib\$6la\$1 at smc.vnet.net...
>
> wolfram who are escalating it to the developers. Possibly someone here has
>
> Integrate[Cos[\[Beta]] Exp[I z Cos[\[Beta]-\[Alpha]]],{\[Beta],0,2
> \[Pi]},Assumptions->z\[Element]Reals]
>
> doesn't seem to run.
>
>
> 2 I \[Pi] BesselJ[1,z] Cos[\[Alpha]]  [ I think ]
>
> thanks
>
>
> Peter Lindsay
>
>

```

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