What is the correct value of $int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x)$?












1












$begingroup$


Integral (6.679.4) in Gradshteyn and Ryzhik claims that
$$
int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x) = frac{2}{pi} sinh(pi b) left[ K_{ib}(a) right]^2
$$
I think that this is not correct. Here is a screenshot of some numerics in Mathematica:



enter image description here



For random values $a=2.34$ and $b=3$, it seems that the LHS of the above evaluates to $0.408$ while the RHS evaluates to $0.653$, and so they disagree.



I think there are three options:



1. I am making a mistake somehow in the above.



2. Mathematica is defining Bessel functions $J$ or $K$ differently than G&R, which I think is not likely.



2. This is an error in G&R.



In the last case, I have seen that Equation (58) on Page 115 of Erdelyi: Volume 1 Tables of Integral Transforms is the source for G&R (6.679.4). This integral simply stated there without a source, so I have no clue how this integral is derived.



What is $int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x)$?






integration







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    A warning: the integral oscillates heavily. In these cases, numerical integration is not always reliable. So, it might be too early to claim an error in G&R.
    $endgroup$
    – Fabian
    Jan 8 at 22:03
















1












$begingroup$


Integral (6.679.4) in Gradshteyn and Ryzhik claims that
$$
int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x) = frac{2}{pi} sinh(pi b) left[ K_{ib}(a) right]^2
$$
I think that this is not correct. Here is a screenshot of some numerics in Mathematica:



enter image description here



For random values $a=2.34$ and $b=3$, it seems that the LHS of the above evaluates to $0.408$ while the RHS evaluates to $0.653$, and so they disagree.



I think there are three options:



1. I am making a mistake somehow in the above.



2. Mathematica is defining Bessel functions $J$ or $K$ differently than G&R, which I think is not likely.



2. This is an error in G&R.



In the last case, I have seen that Equation (58) on Page 115 of Erdelyi: Volume 1 Tables of Integral Transforms is the source for G&R (6.679.4). This integral simply stated there without a source, so I have no clue how this integral is derived.



What is $int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x)$?






integration







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    A warning: the integral oscillates heavily. In these cases, numerical integration is not always reliable. So, it might be too early to claim an error in G&R.
    $endgroup$
    – Fabian
    Jan 8 at 22:03














1












1








1





$begingroup$


Integral (6.679.4) in Gradshteyn and Ryzhik claims that
$$
int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x) = frac{2}{pi} sinh(pi b) left[ K_{ib}(a) right]^2
$$
I think that this is not correct. Here is a screenshot of some numerics in Mathematica:



enter image description here



For random values $a=2.34$ and $b=3$, it seems that the LHS of the above evaluates to $0.408$ while the RHS evaluates to $0.653$, and so they disagree.



I think there are three options:



1. I am making a mistake somehow in the above.



2. Mathematica is defining Bessel functions $J$ or $K$ differently than G&R, which I think is not likely.



2. This is an error in G&R.



In the last case, I have seen that Equation (58) on Page 115 of Erdelyi: Volume 1 Tables of Integral Transforms is the source for G&R (6.679.4). This integral simply stated there without a source, so I have no clue how this integral is derived.



What is $int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x)$?






integration







share|cite|improve this question









$endgroup$




Integral (6.679.4) in Gradshteyn and Ryzhik claims that
$$
int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x) = frac{2}{pi} sinh(pi b) left[ K_{ib}(a) right]^2
$$
I think that this is not correct. Here is a screenshot of some numerics in Mathematica:



enter image description here



For random values $a=2.34$ and $b=3$, it seems that the LHS of the above evaluates to $0.408$ while the RHS evaluates to $0.653$, and so they disagree.



I think there are three options:



1. I am making a mistake somehow in the above.



2. Mathematica is defining Bessel functions $J$ or $K$ differently than G&R, which I think is not likely.



2. This is an error in G&R.



In the last case, I have seen that Equation (58) on Page 115 of Erdelyi: Volume 1 Tables of Integral Transforms is the source for G&R (6.679.4). This integral simply stated there without a source, so I have no clue how this integral is derived.



What is $int_0^infty dx J_0left( 2 a sinhleft( frac{x}{2} right) right) sin(b x)$?






integration




integration






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 8 at 21:33









Greg.PaulGreg.Paul

787921




787921








  • 1




    $begingroup$
    A warning: the integral oscillates heavily. In these cases, numerical integration is not always reliable. So, it might be too early to claim an error in G&R.
    $endgroup$
    – Fabian
    Jan 8 at 22:03














  • 1




    $begingroup$
    A warning: the integral oscillates heavily. In these cases, numerical integration is not always reliable. So, it might be too early to claim an error in G&R.
    $endgroup$
    – Fabian
    Jan 8 at 22:03








1




1




$begingroup$
A warning: the integral oscillates heavily. In these cases, numerical integration is not always reliable. So, it might be too early to claim an error in G&R.
$endgroup$
– Fabian
Jan 8 at 22:03




$begingroup$
A warning: the integral oscillates heavily. In these cases, numerical integration is not always reliable. So, it might be too early to claim an error in G&R.
$endgroup$
– Fabian
Jan 8 at 22:03










2 Answers
2






active

oldest

votes


















2












$begingroup$

Maple agrees with G&R on the formula, however it has considerable difficulty with the numerical evaluation of the integral as this oscillates very rapidly. But the integral from $0$ to $20$ is given as $.6536634068$, agreeing quite nicely with the theoretical result.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
    $endgroup$
    – Greg.Paul
    Jan 9 at 14:14



















1












$begingroup$

Interesting problem, for sure.



Using another CAS, I played with the precision. For Mathematica, your first command would be





f[a_,b_,n_]:=NIntegrate[BesselJ[0, 2 a Sinh[x/2]] Sin[b x], {x, 0, Infinity},
WorkingPrecision -> n]





Below are my results for the integration up to infinity for your particular values.
$$left(
begin{array}{cc}
n & text{result} \
10 & 0.4084616144 \
20 & 0.4084616143 \
30 & 0.6848603276 \
40 & 0.6848603276 \
50 & 0.6554693076 \
60 & 0.6554693076 \
70 & 0.6554693076 \
80 & 0.6554693076 \
90 & 0.6541206739\
100 & 0.6541738396 \
200 & 0.6533938298 \
300 & 0.6534598266 \
400 & 0.6536447955 \
500 & 0.6534985760
end{array}
right)$$ It seems to be very slow convergence (notice the swings of these results even for large values of $n$).



The systematic complain of the CAS was related to the highly oscillatory integrand as Robert Israel commented.






share|cite|improve this answer









$endgroup$














    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066748%2fwhat-is-the-correct-value-of-int-0-infty-dx-j-0-left-2-a-sinh-left-frac%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    Maple agrees with G&R on the formula, however it has considerable difficulty with the numerical evaluation of the integral as this oscillates very rapidly. But the integral from $0$ to $20$ is given as $.6536634068$, agreeing quite nicely with the theoretical result.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
      $endgroup$
      – Greg.Paul
      Jan 9 at 14:14
















    2












    $begingroup$

    Maple agrees with G&R on the formula, however it has considerable difficulty with the numerical evaluation of the integral as this oscillates very rapidly. But the integral from $0$ to $20$ is given as $.6536634068$, agreeing quite nicely with the theoretical result.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
      $endgroup$
      – Greg.Paul
      Jan 9 at 14:14














    2












    2








    2





    $begingroup$

    Maple agrees with G&R on the formula, however it has considerable difficulty with the numerical evaluation of the integral as this oscillates very rapidly. But the integral from $0$ to $20$ is given as $.6536634068$, agreeing quite nicely with the theoretical result.






    share|cite|improve this answer









    $endgroup$



    Maple agrees with G&R on the formula, however it has considerable difficulty with the numerical evaluation of the integral as this oscillates very rapidly. But the integral from $0$ to $20$ is given as $.6536634068$, agreeing quite nicely with the theoretical result.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jan 8 at 22:03









    Robert IsraelRobert Israel

    330k23219473




    330k23219473












    • $begingroup$
      What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
      $endgroup$
      – Greg.Paul
      Jan 9 at 14:14


















    • $begingroup$
      What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
      $endgroup$
      – Greg.Paul
      Jan 9 at 14:14
















    $begingroup$
    What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
    $endgroup$
    – Greg.Paul
    Jan 9 at 14:14




    $begingroup$
    What an awesome and simple trick! I should have known better than to doubt Gradshteyn and Ryzhik :)
    $endgroup$
    – Greg.Paul
    Jan 9 at 14:14











    1












    $begingroup$

    Interesting problem, for sure.



    Using another CAS, I played with the precision. For Mathematica, your first command would be





    f[a_,b_,n_]:=NIntegrate[BesselJ[0, 2 a Sinh[x/2]] Sin[b x], {x, 0, Infinity},
    WorkingPrecision -> n]





    Below are my results for the integration up to infinity for your particular values.
    $$left(
    begin{array}{cc}
    n & text{result} \
    10 & 0.4084616144 \
    20 & 0.4084616143 \
    30 & 0.6848603276 \
    40 & 0.6848603276 \
    50 & 0.6554693076 \
    60 & 0.6554693076 \
    70 & 0.6554693076 \
    80 & 0.6554693076 \
    90 & 0.6541206739\
    100 & 0.6541738396 \
    200 & 0.6533938298 \
    300 & 0.6534598266 \
    400 & 0.6536447955 \
    500 & 0.6534985760
    end{array}
    right)$$     It seems to be very slow convergence (notice the swings of these results even for large values of $n$).



    The systematic complain of the CAS was related to the highly oscillatory integrand as Robert Israel commented.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Interesting problem, for sure.



      Using another CAS, I played with the precision. For Mathematica, your first command would be





      f[a_,b_,n_]:=NIntegrate[BesselJ[0, 2 a Sinh[x/2]] Sin[b x], {x, 0, Infinity},
      WorkingPrecision -> n]





      Below are my results for the integration up to infinity for your particular values.
      $$left(
      begin{array}{cc}
      n & text{result} \
      10 & 0.4084616144 \
      20 & 0.4084616143 \
      30 & 0.6848603276 \
      40 & 0.6848603276 \
      50 & 0.6554693076 \
      60 & 0.6554693076 \
      70 & 0.6554693076 \
      80 & 0.6554693076 \
      90 & 0.6541206739\
      100 & 0.6541738396 \
      200 & 0.6533938298 \
      300 & 0.6534598266 \
      400 & 0.6536447955 \
      500 & 0.6534985760
      end{array}
      right)$$       It seems to be very slow convergence (notice the swings of these results even for large values of $n$).



      The systematic complain of the CAS was related to the highly oscillatory integrand as Robert Israel commented.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Interesting problem, for sure.



        Using another CAS, I played with the precision. For Mathematica, your first command would be





        f[a_,b_,n_]:=NIntegrate[BesselJ[0, 2 a Sinh[x/2]] Sin[b x], {x, 0, Infinity},
        WorkingPrecision -> n]





        Below are my results for the integration up to infinity for your particular values.
        $$left(
        begin{array}{cc}
        n & text{result} \
        10 & 0.4084616144 \
        20 & 0.4084616143 \
        30 & 0.6848603276 \
        40 & 0.6848603276 \
        50 & 0.6554693076 \
        60 & 0.6554693076 \
        70 & 0.6554693076 \
        80 & 0.6554693076 \
        90 & 0.6541206739\
        100 & 0.6541738396 \
        200 & 0.6533938298 \
        300 & 0.6534598266 \
        400 & 0.6536447955 \
        500 & 0.6534985760
        end{array}
        right)$$         It seems to be very slow convergence (notice the swings of these results even for large values of $n$).



        The systematic complain of the CAS was related to the highly oscillatory integrand as Robert Israel commented.






        share|cite|improve this answer









        $endgroup$



        Interesting problem, for sure.



        Using another CAS, I played with the precision. For Mathematica, your first command would be





        f[a_,b_,n_]:=NIntegrate[BesselJ[0, 2 a Sinh[x/2]] Sin[b x], {x, 0, Infinity},
        WorkingPrecision -> n]





        Below are my results for the integration up to infinity for your particular values.
        $$left(
        begin{array}{cc}
        n & text{result} \
        10 & 0.4084616144 \
        20 & 0.4084616143 \
        30 & 0.6848603276 \
        40 & 0.6848603276 \
        50 & 0.6554693076 \
        60 & 0.6554693076 \
        70 & 0.6554693076 \
        80 & 0.6554693076 \
        90 & 0.6541206739\
        100 & 0.6541738396 \
        200 & 0.6533938298 \
        300 & 0.6534598266 \
        400 & 0.6536447955 \
        500 & 0.6534985760
        end{array}
        right)$$         It seems to be very slow convergence (notice the swings of these results even for large values of $n$).



        The systematic complain of the CAS was related to the highly oscillatory integrand as Robert Israel commented.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 9 at 7:06









        Claude LeiboviciClaude Leibovici

        125k1158136




        125k1158136






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066748%2fwhat-is-the-correct-value-of-int-0-infty-dx-j-0-left-2-a-sinh-left-frac%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Måne

            Image
            För jordens måne, se Månen. Några månar i solsystemet i skala till vår måne En måne , naturlig satellit eller drabant är en himlakropp som kretsar kring en planet eller en asteroid i ett solsystem. Ordet i bestämd form – månen – avser planeten jordens måne. Det finns inte i något fall belagt att någon måne har egna månar. De flesta stora månar i solsystemet är bundna i synkron rotation med en och samma sida vänd mot planeten. Ett exempel på en måne som inte beter sig på detta vis, är Hyperion som kaotiskt roterar kring Saturnus. En asteroidmåne är en måne som kretsar kring en asteroid. En måne kan ha bildats samtidigt som moderplaneten eller senare infångats av planetens gravitationskraft. I vårt eget solsystem känner vi i nuläget till 173 månar runt sex av de åtta planeterna [ 1 ] : Jorden – månen Mars – Phobos och Deimos Jupiter – Ganymedes (som är solsystemets största måne), Io, Europa, Callisto och 59 andra kända månar Saturnus – Titan (den enda måne...
            Read more

            Storängen

            Image
            För stadsdelen i Mariehamn, se Storängen, Åland. För ett naturreservat i Norrtälje kommun, se Storängens naturreservat. Storängen från luften i september 1967, vy mot väst. Till vänster syns Järlasjön och Saltsjöbanans spår. Till höger löper Värmdövägen och längst ut till höger syns anläggningsarbetena för Värmdöleden. Storängen är ett villaområde i centrala Nacka i Stockholms län. Området är beläget strax sydost om Nacka Forum vid Saltsjöbanan mellan Lillängen och Saltsjö-Duvnäs. Storängens villasamhälle grundades 1904 av "Tjänstemännens Egnahemsförening vid Storängen" och de första villorna började byggas samma år. Redan 1909 var 107 tomter bebyggda av totalt runt 160. [ 1 ] År 2013 fanns drygt 150 villor med lika många familjer i området. Egnahemsföreningen existerar fortfarande och fungerar idag som intresseförening för de boende i Storängen. Storängen anses vara ett av Sveriges bäst bevarade villasamhällen från 1900-talets början, tack vare sina tidstypis...
            Read more

            VLT Carioca

            Image
            VLT Carioca Informações Proprietário Prefeitura do Rio de Janeiro Local [[Rio de Janeiro, RJ]] País {{country data Brasil| country flag2| name = Brasil| variant = | size = }} Tipo de transporte Bonde (Veículo leve sobre trilhos) Número de linhas 2 (+1 em construção) Número de estações 26 (+3 em construção) Tráfego 80 mil passageiros/dia [ 1 ] (dezembro/2018) Website www.vltrio.com.br Funcionamento Início de funcionamento 5 de junho de 2016 (2 anos) Operadora(s) Concessionária do VLT Carioca S.A. Número de veículos 32 Alstom Citadis Comprimento dos veículos 44 m (144 ft) Dados técnicos Extensão do sistema 28 km (17,4 mi) [ 2 ] Frequência 5–10 minutos Bitola Bitola padrão/internacional 1 435 mm (4,71 ft) Eletrificação APS, 750V DC Velocidade média 15 km/h (9,32 mph) [ 2 ] Velocidade máxima 50 km/h (31,1 mph) ...
            Read more