Re: an integral containing BesselJ
- To: mathgroup at smc.vnet.net
- Subject: [mg67657] Re: an integral containing BesselJ
- From: dimmechan at yahoo.com
- Date: Mon, 3 Jul 2006 06:38:46 -0400 (EDT)
- References: <e784gb$fph$1@smc.vnet.net><e7tehh$438$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Î?/Î? ab_def at prontomail.com ÎγÏ?αÏ?ε:
> dimmechan at yahoo.com wrote:
> > In Mathematica 5.2 i took the result:
> >
> > In[1]:=
> > \!\(Integrate[\(Log[x]\/\@\(1 + x\^2\)\) BesselJ[0, x], {x, 0, 8}] //
> > Timing\)
> >
> > Out[1]=
> > {46.171 Second,0}
> >
> > which is a special case of:
> >
> > In[2]:=
> > \!\(Integrate[\(Log[a\ x]\/\@\(1 + x\^2\)\) BesselJ[0,
> > x], {x, 0, 8}] // Timing\)
> >
> > Out[2]=
> > \!\({32.18800000000001`\ Second, BesselI[0, 1\/2]\ BesselK[0, 1\/2]\
> > Log[a]}\)
> >
> > However performing the numerical integration gave the result:
> >
> > In[4]:=
> > \!\(NIntegrate[\(Log[x]\/\@\(1 +
> > x\^2\)\) BesselJ[0, x], {x, 0, 8}, Method \[Rule]
> > Oscillatory] // Timing\)
> >
> > Out[4]=
> > {0.75 Second,-0.997939}
> >
> > I also perfrorm the symbolic integration in Mathematica 4.0 and took a
> > lengthy result, which agrees numerically with the obtained result from
> > the numerical integration:
> >
> > In[1]:=
> > \!\(Integrate[\(Log[a\ x]\/\@\(1 + x\^2\)\) BesselJ[0, x], {x,
> > 0, \[Infinity]}] // Timing\)
> >
> > Out[1]=
> > \!\(\*
> > RowBox[{"{",
> > RowBox[{\(18.594`\ Second\), ",",
> > RowBox[{\(1\/12\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1},
> > 1\/4])\)\ \((6\ EulerGamma\^2 - \[Pi]\^2 -
> > 6\ EulerGamma\ Log[4])\)\),
> > "+", \(1\/12\ \((6\ EulerGamma\^2 - \[Pi]\^2 -
> > 6\ EulerGamma\ Log[4] +
> > 12\ BesselI[0, 1\/2]\ BesselK[0, 1\/2]\ Log[a])\)\), "+",
> >
> > RowBox[{\(1\/\(2\ \@\[Pi]\)\),
> > RowBox[{"(",
> > RowBox[{\(Log[4]\), " ",
> > RowBox[{"(",
> >
> > RowBox[{\(1\/2\ \@\[Pi]\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1}, 1\/4])\)\
> > Log[4]\),
> > "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14174 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(2\^\(\(-1\) - 2\ K$14174\)\ Gamma[
> > 1\/2 + K$14174]\ PolyGamma[0,
> > 1\/2 + K$14174]\)\/Gamma[1 +
> > K$14174]\^3\)}], "+",
> >
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14174 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(-\(\(2\^\(\(-1\) - 2\ K$14174\)\ Gamma[
> > 1\/2 + K$14174]\ PolyGamma[0,
> > 1 + K$14174]\)\/Gamma[1 +
> > K$14174]\^3\)\)}]}],
> > ")"}]}], ")"}]}], "+",
> > FractionBox[
> > RowBox[{\((\(-2\)\ EulerGamma + Log[4])\), " ",
> > RowBox[{"(",
> >
> > RowBox[{\(EulerGamma\ \@\[Pi]\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1}, 1\/4])\)\),
> > "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14447 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(4\^\(-K$14447\)\ Gamma[
> > 1\/2 + K$14447]\ PolyGamma[0,
> > 1 + K$14447]\)\/Gamma[1 + K$14447]\^3\)}]}],
> > ")"}]}], \(2\ \@\[Pi]\)], "+",
> > RowBox[{\(1\/\(3\ \@\[Pi]\)\),
> > RowBox[{"(",
> > RowBox[{\(1\/8\ \[Pi]\^\(5/2\)\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1}, 1\/4])\)\),
> > "-", \(3\/8\ \@\[Pi]\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1}, 1\/4])\)\
> > Log[4]\^2\),
> > "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(-\(\(3\ 2\^\(\(-3\) - 2\ K$14601\)\ Gamma[
> > 1\/2 + K$14601]\ Log[16]\ PolyGamma[0,
> > 1\/2 + K$14601]\)\/Gamma[1 +
> > K$14601]\^3\)\)}], "+",
> >
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(-\(\(3\ 2\^\(\(-3\) - 2\ K$14601\)\ Gamma[
> > 1\/2 +
> > K$14601]\ PolyGamma[0, 1\/2 + \
> > K$14601]\^2\)\/Gamma[1 + K$14601]\^3\)\)}], "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(3\ 2\^\(\(-3\) - 2\ K$14601\)\ Gamma[
> > 1\/2 + K$14601]\ Log[16]\ PolyGamma[0,
> > 1 + K$14601]\)\/Gamma[1 + K$14601]\^3\)}], "+",
> >
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(3\ 2\^\(\(-2\) - 2\ K$14601\)\ Gamma[
> > 1\/2 + K$14601]\ PolyGamma[0,
> > 1\/2 + K$14601]\ PolyGamma[0,
> > 1 + K$14601]\)\/Gamma[1 + K$14601]\^3\)}], "+",
> >
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(-\(\(3\ 2\^\(\(-3\) - 2\ K$14601\)\ Gamma[
> > 1\/2 +
> > K$14601]\ PolyGamma[0, 1 +
> > K$14601]\^2\)\/Gamma[1 \
> > + K$14601]\^3\)\)}], "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(-\(\(3\ 2\^\(\(-3\) - 2\ K$14601\)\ Gamma[
> > 1\/2 + K$14601]\ PolyGamma[1,
> > 1\/2 + K$14601]\)\/Gamma[1 +
> > K$14601]\^3\)\)}], "+",
> >
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$14601 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(3\ 2\^\(\(-3\) - 2\ K$14601\)\ Gamma[
> > 1\/2 + K$14601]\ PolyGamma[1,
> > 1 + K$14601]\)\/Gamma[1 + K$14601]\^3\)}]}],
> > ")"}]}],
> > "+",
> > RowBox[{\(1\/\(3\ \@\[Pi]\)\),
> > RowBox[{"(",
> >
> > RowBox[{\(3\/2\ EulerGamma\^2\ \@\[Pi]\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1}, 1\/4])\)\),
> > "+", \(1\/8\ \[Pi]\^\(5/2\)\ \((\(-1\) +
> > HypergeometricPFQ[{1\/2}, {1, 1}, 1\/4])\)\), "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$15304 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(3\ 4\^\(-K$15304\)\ EulerGamma\ Gamma[
> > 1\/2 + K$15304]\ PolyGamma[0,
> > 1 + K$15304]\)\/Gamma[1 + K$15304]\^3\)}], "+",
> >
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$15304 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(\(3\ 2\^\(\(-1\) - 2\ K$15304\)\ Gamma[
> > 1\/2 +
> > K$15304]\ PolyGamma[0, 1 +
> > K$15304]\^2\)\/Gamma[1 + \
> > K$15304]\^3\)}], "+",
> > RowBox[{
> > UnderoverscriptBox["\[Sum]", \(K$15304 = 1\),
> > InterpretationBox["\[Infinity]",
> > DirectedInfinity[
> > 1]]], \(-\(\(3\ 2\^\(\(-2\) - 2\ K$15304\)\ Gamma[
> > 1\/2 + K$15304]\ PolyGamma[1,
> > 1 + K$15304]\)\/Gamma[1 +
> > K$15304]\^3\)\)}]}],
> > ")"}]}]}]}], "}"}]\)
> >
> > In[2]:=
> > N[%[[2]]]
> >
> > Out[2]=
> > 0.0580341\[InvisibleSpace] + 0.0833333 (-12.6717 + 11.7973 Log[a])
> >
> > In[3]:=
> > % /. a -> 1
> >
> > Out[3]=
> > -0.997939
> >
> > Can anyone help me?
>
> Here's a link to a paper that explains how such integrals can be
> evaluated: http://www.cs.cmu.edu/~adamchik/articles/integr/mier.nb
> (also have a look at issac90.pdf). First we find the transforms of the
> factors:
>
> In[1]:= Integrate[Log[1/x]/Sqrt[1 + 1/x^2]*x^(s - 1), {x, 0, Infinity}]
>
> In[2]:= Integrate[BesselJ[0, x]*x*x^(s - 1), {x, 0, Infinity}]
>
> In[3]:= fs = Simplify[%%*%, -1 < s < 0]
>
> Out[3]= (2^(-2 + s)*Gamma[-(s/2)]*Gamma[(1 + s)/2]^2*(PolyGamma[0,
> -(s/2)] - PolyGamma[0, (1 + s)/2]))/(Sqrt[Pi]*Gamma[1/2 - s/2])
>
> The original integral is equal to 1/(2*Pi*I) times the integral of fs
> along the line Re[s] == -1/2:
>
> In[4]:= 1/(2*Pi*I)*NIntegrate[fs, {s, -I*Infinity, -1/2, I*Infinity}]
>
> Out[4]= -0.9979393746397847 - 2.6504622345529306*^-17*I
>
> We can represent fs as a sum of the derivatives of gamma function
> ratios:
>
> In[5]:= fs == D[2^(-2 + s)*Gamma[-s/2 + a]*Gamma[(1 + s)/2]^2/
> (Sqrt[Pi]*Gamma[(1 - s)/2]) -
> 1/2*2^(-2 + s)*Gamma[-s/2]*Gamma[(1 + s)/2 + a]^2/
> (Sqrt[Pi]*Gamma[(1 - s)/2]),
> a] /. a -> 0 // Simplify
>
> Out[5]= True
>
> Since the integral of the ratio of gamma functions is a Meijer
> function, the answer is expressible in terms of the derivatives of
> MeijerG:
>
> In[6]:= D[1/(4*Sqrt[Pi])*MeijerG[{{1/2 - a}, {}}, {{0, 0}, {0}}, 1/4] -
> 1/(8*Sqrt[Pi])*MeijerG[{{1/2}, {}}, {{a, a}, {0}}, 1/4],
> a] /. a -> 0 // Simplify
>
> Out[6]= (1/(8*Sqrt[Pi]))*(-Derivative[{{0}, {}}, {{0, 1}, {0}},
> 0][MeijerG][{{1/2}, {}}, {{0, 0}, {0}}, 1/4] - Derivative[{{0}, {}},
> {{1, 0}, {0}}, 0][MeijerG][{{1/2}, {}}, {{0, 0}, {0}}, 1/4] -
> 2*Derivative[{{1}, {}}, {{0, 0}, {0}}, 0][MeijerG][{{1/2}, {}}, {{0,
> 0}, {0}}, 1/4])
>
> This is probably not very useful, because Mathematica cannot evaluate
> N[%6], and FunctionExpand[%6] gives Indeterminate.
>
> Using the same technique of differentiation with respect to a
> parameter, we can express the integral in terms of the derivatives of
> HypergeometricPFQ:
>
> In[7]:= int = (1/24)*(-2*BesselI[0, 1/2]^2*(Pi^2 +
> 6*EulerGamma*(-EulerGamma + Log[4])) + (9*EulerGamma -
> 3*Log[16])*Derivative[{0}, {0, 1}, 0][HypergeometricPFQ][{1/2}, {1, 1},
> 1/4] + 3*Log[16]*Derivative[{1}, {0, 0}, 0][HypergeometricPFQ][{1/2},
> {1, 1}, 1/4] + 3*Derivative[{1}, {0, 1}, 0][HypergeometricPFQ][{1/2},
> {1, 1}, 1/4] - 3*Derivative[{2}, {0, 0}, 0][HypergeometricPFQ][{1/2},
> {1, 1}, 1/4] - 15*EulerGamma*Derivative[{0, 1}, {0, 0, 0},
> 0][HypergeometricPFQ][{1/2, 1}, {1, 1, 1}, 1/4] - 12*Derivative[{0, 1},
> {0, 0, 1}, 0][HypergeometricPFQ][{1/2, 1}, {1, 1, 1}, 1/4] -
> 3*Derivative[{0, 2}, {0, 0, 0}, 0][HypergeometricPFQ][{1/2, 1}, {1, 1,
> 1}, 1/4] + 9*Derivative[{1, 1}, {0, 0, 0}, 0][HypergeometricPFQ][{1/2,
> 1}, {1, 1, 1}, 1/4]);
>
> Mathematica seems to get the approximations for mixed partial
> derivatives wrong:
>
> In[8]:= N[Derivative[{1, 1}, {0, 0, 0}, 0][
> HypergeometricPFQ][{1/2, 1}, {1, 1, 1}, 1/4],
> 20]
>
> Out[8]= 4.523987519373309478408412473406`20.*^40
>
> This looks like a bug, because even the simplest finite difference
> formula can give a good approximation:
>
> In[9]:= Block[{$MaxExtraPrecision = 100},
> int /.
> Derivative[{si: (0)..., 1, si2: (0)..., 1, si3: (0)...},
> {(0)..}, 0][HypergeometricPFQ][a_, b_, z_] :>
> ((f[h, k] - f[h, -k] - f[-h, k] + f[-h, -k])/(4*h*k) /.
> f -> Function @@ {HypergeometricPFQ[
> a + {si, #, si2, #2, si3}, b, z]}) /.
> {h -> 10^-25, k -> 10^-25} //
> N[#, 50]&
> ]
>
> Out[9]=
> -0.9979393746360745614717131506926624475792306906555576417821764083894`50.
>
> Generalizing to Log[a*x] is easy:
>
> Integrate[Log[a*x]/Sqrt[1 + x^2]*BesselJ[0, x], {x, 0, Infinity}] ==
>
> Integrate[(Log[x] + Log[a])/Sqrt[1 + x^2]*BesselJ[0, x],
>
> {x, 0, Infinity}] ==
>
> int + BesselI[0, 1/2]*BesselK[0, 1/2]*Log[a]
>
> Maxim Rytin
> m.r at inbox.ru
Thank you for your respose