Ward identities for 1D integrals











up vote
2
down vote

favorite












Some integrals of type



$$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$



can be solved by realizing that they satisfy a differential equation
like



$$ left(a - d^2/da^2right)I(a)=0,$$



and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
but the same can be done for more general integrands of the similar form.



My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
i.e. something like



$$ I(a_1, ldots, a_n)
= int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$



Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?






integration differential-equations







share|cite|improve this question


























    up vote
    2
    down vote

    favorite












    Some integrals of type



    $$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$



    can be solved by realizing that they satisfy a differential equation
    like



    $$ left(a - d^2/da^2right)I(a)=0,$$



    and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
    but the same can be done for more general integrands of the similar form.



    My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
    i.e. something like



    $$ I(a_1, ldots, a_n)
    = int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$



    Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?






    integration differential-equations







    share|cite|improve this question
























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Some integrals of type



      $$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$



      can be solved by realizing that they satisfy a differential equation
      like



      $$ left(a - d^2/da^2right)I(a)=0,$$



      and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
      but the same can be done for more general integrands of the similar form.



      My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
      i.e. something like



      $$ I(a_1, ldots, a_n)
      = int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$



      Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?






      integration differential-equations







      share|cite|improve this question













      Some integrals of type



      $$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$



      can be solved by realizing that they satisfy a differential equation
      like



      $$ left(a - d^2/da^2right)I(a)=0,$$



      and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
      but the same can be done for more general integrands of the similar form.



      My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
      i.e. something like



      $$ I(a_1, ldots, a_n)
      = int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$



      Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?






      integration differential-equations




      integration differential-equations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked yesterday









      z.v.

      189111




      189111



























          active

          oldest

          votes











          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',
          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%2f3005037%2fward-identities-for-1d-integrals%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown






























          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















           

          draft saved


          draft discarded



















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005037%2fward-identities-for-1d-integrals%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

          Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

          Image
          28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
          Read more

          Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

          Image
          up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
          Read more

          A Topological Invariant for $pi_3(U(n))$

          Image
          3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
          Read more