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MathGroup Archive 2006

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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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