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?

```

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