Re: Re: Re: A Bessel integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg36914] Re: [mg36887] Re: Re: A Bessel integral*From*: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>*Date*: Wed, 2 Oct 2002 03:31:38 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Your Mathematica 4.2 is certainly not like the one most of us have: > This reveals us another integral which Mathematica 4.1 fails to > calculate > > In[1] := Integrate[Cos[Pi/4 - z]^2/z, {z, 1, Infinity}] // N > Out[1]= -0.0173083 > > In[2] := $Version > Out[2]= "4.1 for Microsoft Windows (November 2, 2000)" > > but Mathematica 4.2 handles correctly > > In[1] := Integrate[Cos[Pi/4 - z]^2/z, {z, 1, Infinity}] > Out[1] = Integrate::"idiv": "Integral of"... "does not converge on > {1, Infinity)." > > In[2] := $Version > Out[2]= "4.2 for Microsoft Windows (February 28, 2002)" Well, actually with my 4.2 we get: In[1]:= Integrate[Cos[Pi/4 - z]^2/z, {z, 1, Infinity}] Out[1]= (1/4)*(Pi - 2*SinIntegral[2]) > > > Even simpler, > > In[1] := Integrate[Cos[z]^2/z, {z, 1, Infinity}] > Out[1] = -EulerGamma/2 - Log[2]/2 + (EulerGamma - CosIntegral[2] + > Log[2])/2 > > In[2] := $Version > Out[2]= "4.1 for Microsoft Windows (November 2, 2000)" > > which is wrong while Mathematica 4.2 works excellent > > In[1] := Integrate[Cos[z]^2/z, {z, 1, Infinity}] > Out[1] = Integrate::"idiv": "Integral of"..."does not converge on > {1, Infinity)." > > In[2] := $Version > Out[2]= "4.2 for Microsoft Windows (February 28, 2002)" Not so fast: In[2]:= Integrate[Cos[z]^2/z, {z, 1, Infinity}] Out[2]= -(EulerGamma/2) - Log[2]/2 + (1/2)*(EulerGamma - CosIntegral[2] + Log[2]) I'd speculate that fixing these two integrals in the beta stage some more important ones, so the original way of doing things was restored in the released version. However, it gets even more interesting if we load: << Calculus`Limit` In[4]:= Integrate[Cos[Pi/4-z]^2/z,{z,1,Infinity}] ReplaceRepeated:: rrlim :"Exiting after Interval[{-1,1}]/z + Interval[{0, 1}]/z scanned 65536 times." From In[4]:= Integrate:: idiv :Integral of Cos[Pi]/4 - z^2/z does not converge on {1, Infinity]. Out[4]= Integrate[Cos[Pi/4 - z]^2/z, {z, 1, Infinity}] In[5]:= Integrate[Cos[z]^2/z, {z, 1, Infinity}] From In[5]:= ReplaceRepeated::rrlim:Exiting after Interval[{0,1}] scanned 65536 times. From In[5]:= Integrate::idiv:Integral of Cos[z]^2/z does not converge on {1, Infinity} Out[5]= Integrate[Cos[z]^2/z, {z, 1, Infinity}] In[6]:= $Version Out[6]= 4.2 for Mac OS X (June 4, 2002) > > > Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ http://platon.c.u-tokyo.ac.jp/andrzej/