       Re: troublesome integral

• To: mathgroup at smc.vnet.net
• Subject: [mg126010] Re: troublesome integral
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Fri, 13 Apr 2012 04:44:21 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jm64ib\$6la\$1@smc.vnet.net>

```On 12 Apr., 10:42, peter lindsay <p... at me.com> wrote:
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]] =C2 [ I think ]
>
> thanks
>
> Peter Lindsay
>
>
As it happens frequently, Mathematica likes to accept some help:

1) let b-a=g and consider

In:=
FullSimplify[Integrate[Cos[g - a]*Exp[I*z*Cos[g]],
{g, 0, 2*Pi}], z =E2=88=88 Reals]

Out=
2*I*Pi*BesselJ[1, z]*Cos[a]

Here an error has crept in: the limits of integration are not correct,

2) The result is nevertheless correct as can be seen with the original
integral by expanding the exponential function, integrating term by
term, summing up

In:=
Distribute[
Plus @@ Table[Integrate[Cos[b]*(I*z*Cos[b - a])^k,
{b, 0, 2*Pi}, Assumptions -> {z =E2=88=88 Reals,
k =E2=88=88 Integers}]/k!, {k, 0, 10}]/
(I*Pi*Cos[a])]

Out=
z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280

and comparing this to the suspected result

In:=
Normal[2*Series[BesselJ[1, z], {z, 0, 10}]]

Out=
z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280

No stict proof as 10 != inf but very plausible.

Regards,
Wolfgang

```

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