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: > A couple of colleagues wondered about this. I've sent it on to support @ wolfram who are escalating it to the developers. Possibly someone here has an answer though ? > > Integrate[Cos[\[Beta]] Exp[I z Cos[\[Beta]-\[Alpha]]],{\[Beta],0,2 \[Pi]}= ,Assumptions->z\[Element]Reals] > > doesn't seem to run. > > Answer should be > > 2 I \[Pi] BesselJ[1,z] Cos[\[Alpha]] =C2 [ I think ] > > thanks > > Peter Lindsay > > Your result is correct. As it happens frequently, Mathematica likes to accept some help: 1) let b-a=g and consider In[44]:= FullSimplify[Integrate[Cos[g - a]*Exp[I*z*Cos[g]], {g, 0, 2*Pi}], z =E2=88=88 Reals] Out[44]= 2*I*Pi*BesselJ[1, z]*Cos[a] Here an error has crept in: the limits of integration are not correct, they should read {g,-a,2 Pi-a} 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[48]:= 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[48]= z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280 and comparing this to the suspected result In[45]:= Normal[2*Series[BesselJ[1, z], {z, 0, 10}]] Out[45]= z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280 No stict proof as 10 != inf but very plausible. Regards, Wolfgang