an integral containing BesselJ
- To: mathgroup at smc.vnet.net
- Subject: [mg67365] an integral containing BesselJ
- From: dimmechan at yahoo.com
- Date: Tue, 20 Jun 2006 02:14:49 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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?