How to calculate contour integration of multivalued functions?












1












$begingroup$


enter image description here



I have a question about contour integral of the multivalude function. The question is from paper arXiv:1211.6767 (page 6-7).



I want to calculate the Fourier transformation of a muti-valued function $f$
$$G(omega,mathbf{k})=int dt d^{d-1}mathbf{x} f(t,mathbf{x})
e^{iomega t-imathbf{k}cdotmathbf{x}}$$
with condition that
$$omega>0,,,,,,,,omega^2-mathbf{k}^2>0$$
so that we can deform the contour from the green one to the red one.
Function $f$ is given by
$$ f(t,mathbf{x})=
big(frac{-1}{t^2-mathbf{x}^2-iepsilon t}big)^Delta
$$
For $t<0$, $f$ can be written as
$$ f(t,mathbf{x})=
frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
$$



For $t>0$, $f$ can be written as
$$ f(t,mathbf{x})=
frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
$$



First I need to write down the function on the four legs, $1,2,3,4$
On the leg 1, since $t<0$, we should have



$$ f_1(t,mathbf{x})=
frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
$$
From leg 1 to leg 2, we need
$$(t+|x|)rightarrow(t+|x|)e^{2pi i} $$
$$ f_2(t,mathbf{x})=
frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
$$
On the leg 4, since $t>0$, we should have



$$ f_4(t,mathbf{x})=
frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
$$
From leg 4 to leg 3, we need
$$(t-|x|)rightarrow(t+|x|)e^{-2pi i} $$



$$ f_3(t,mathbf{x})=
frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
$$



Combining the above result, we have
$$G(omega,mathbf{k})=(e^{-i pi Delta}-e^{+i pi Delta})
int d^{d-1}mathbf{x} int^infty_{|x|} dt
frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} \
+(e^{+i pi Delta}-e^{-i pi Delta})
int d^{d-1}mathbf{x} int^infty_{|x|} dt
frac{e^{-iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta}$$



But the authors' result is quite different
$$G(omega,mathbf{k})=2(e^{i pi Delta}-e^{-i pi Delta})
int d^{d-1}mathbf{x} int^infty_{|x|} dt
frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} $$






complex-analysis







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    enter image description here



    I have a question about contour integral of the multivalude function. The question is from paper arXiv:1211.6767 (page 6-7).



    I want to calculate the Fourier transformation of a muti-valued function $f$
    $$G(omega,mathbf{k})=int dt d^{d-1}mathbf{x} f(t,mathbf{x})
    e^{iomega t-imathbf{k}cdotmathbf{x}}$$
    with condition that
    $$omega>0,,,,,,,,omega^2-mathbf{k}^2>0$$
    so that we can deform the contour from the green one to the red one.
    Function $f$ is given by
    $$ f(t,mathbf{x})=
    big(frac{-1}{t^2-mathbf{x}^2-iepsilon t}big)^Delta
    $$
    For $t<0$, $f$ can be written as
    $$ f(t,mathbf{x})=
    frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
    $$



    For $t>0$, $f$ can be written as
    $$ f(t,mathbf{x})=
    frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
    $$



    First I need to write down the function on the four legs, $1,2,3,4$
    On the leg 1, since $t<0$, we should have



    $$ f_1(t,mathbf{x})=
    frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
    $$
    From leg 1 to leg 2, we need
    $$(t+|x|)rightarrow(t+|x|)e^{2pi i} $$
    $$ f_2(t,mathbf{x})=
    frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
    $$
    On the leg 4, since $t>0$, we should have



    $$ f_4(t,mathbf{x})=
    frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
    $$
    From leg 4 to leg 3, we need
    $$(t-|x|)rightarrow(t+|x|)e^{-2pi i} $$



    $$ f_3(t,mathbf{x})=
    frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
    $$



    Combining the above result, we have
    $$G(omega,mathbf{k})=(e^{-i pi Delta}-e^{+i pi Delta})
    int d^{d-1}mathbf{x} int^infty_{|x|} dt
    frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} \
    +(e^{+i pi Delta}-e^{-i pi Delta})
    int d^{d-1}mathbf{x} int^infty_{|x|} dt
    frac{e^{-iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta}$$



    But the authors' result is quite different
    $$G(omega,mathbf{k})=2(e^{i pi Delta}-e^{-i pi Delta})
    int d^{d-1}mathbf{x} int^infty_{|x|} dt
    frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} $$






    complex-analysis







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      enter image description here



      I have a question about contour integral of the multivalude function. The question is from paper arXiv:1211.6767 (page 6-7).



      I want to calculate the Fourier transformation of a muti-valued function $f$
      $$G(omega,mathbf{k})=int dt d^{d-1}mathbf{x} f(t,mathbf{x})
      e^{iomega t-imathbf{k}cdotmathbf{x}}$$
      with condition that
      $$omega>0,,,,,,,,omega^2-mathbf{k}^2>0$$
      so that we can deform the contour from the green one to the red one.
      Function $f$ is given by
      $$ f(t,mathbf{x})=
      big(frac{-1}{t^2-mathbf{x}^2-iepsilon t}big)^Delta
      $$
      For $t<0$, $f$ can be written as
      $$ f(t,mathbf{x})=
      frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$



      For $t>0$, $f$ can be written as
      $$ f(t,mathbf{x})=
      frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$



      First I need to write down the function on the four legs, $1,2,3,4$
      On the leg 1, since $t<0$, we should have



      $$ f_1(t,mathbf{x})=
      frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$
      From leg 1 to leg 2, we need
      $$(t+|x|)rightarrow(t+|x|)e^{2pi i} $$
      $$ f_2(t,mathbf{x})=
      frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$
      On the leg 4, since $t>0$, we should have



      $$ f_4(t,mathbf{x})=
      frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$
      From leg 4 to leg 3, we need
      $$(t-|x|)rightarrow(t+|x|)e^{-2pi i} $$



      $$ f_3(t,mathbf{x})=
      frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$



      Combining the above result, we have
      $$G(omega,mathbf{k})=(e^{-i pi Delta}-e^{+i pi Delta})
      int d^{d-1}mathbf{x} int^infty_{|x|} dt
      frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} \
      +(e^{+i pi Delta}-e^{-i pi Delta})
      int d^{d-1}mathbf{x} int^infty_{|x|} dt
      frac{e^{-iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta}$$



      But the authors' result is quite different
      $$G(omega,mathbf{k})=2(e^{i pi Delta}-e^{-i pi Delta})
      int d^{d-1}mathbf{x} int^infty_{|x|} dt
      frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} $$






      complex-analysis







      share|cite|improve this question











      $endgroup$




      enter image description here



      I have a question about contour integral of the multivalude function. The question is from paper arXiv:1211.6767 (page 6-7).



      I want to calculate the Fourier transformation of a muti-valued function $f$
      $$G(omega,mathbf{k})=int dt d^{d-1}mathbf{x} f(t,mathbf{x})
      e^{iomega t-imathbf{k}cdotmathbf{x}}$$
      with condition that
      $$omega>0,,,,,,,,omega^2-mathbf{k}^2>0$$
      so that we can deform the contour from the green one to the red one.
      Function $f$ is given by
      $$ f(t,mathbf{x})=
      big(frac{-1}{t^2-mathbf{x}^2-iepsilon t}big)^Delta
      $$
      For $t<0$, $f$ can be written as
      $$ f(t,mathbf{x})=
      frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$



      For $t>0$, $f$ can be written as
      $$ f(t,mathbf{x})=
      frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$



      First I need to write down the function on the four legs, $1,2,3,4$
      On the leg 1, since $t<0$, we should have



      $$ f_1(t,mathbf{x})=
      frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$
      From leg 1 to leg 2, we need
      $$(t+|x|)rightarrow(t+|x|)e^{2pi i} $$
      $$ f_2(t,mathbf{x})=
      frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$
      On the leg 4, since $t>0$, we should have



      $$ f_4(t,mathbf{x})=
      frac{e^{-ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$
      From leg 4 to leg 3, we need
      $$(t-|x|)rightarrow(t+|x|)e^{-2pi i} $$



      $$ f_3(t,mathbf{x})=
      frac{e^{ipi Delta}}{ big(t^2-x^2big)^Delta}
      $$



      Combining the above result, we have
      $$G(omega,mathbf{k})=(e^{-i pi Delta}-e^{+i pi Delta})
      int d^{d-1}mathbf{x} int^infty_{|x|} dt
      frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} \
      +(e^{+i pi Delta}-e^{-i pi Delta})
      int d^{d-1}mathbf{x} int^infty_{|x|} dt
      frac{e^{-iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta}$$



      But the authors' result is quite different
      $$G(omega,mathbf{k})=2(e^{i pi Delta}-e^{-i pi Delta})
      int d^{d-1}mathbf{x} int^infty_{|x|} dt
      frac{e^{iomega t-imathbf{k}cdotmathbf{x}} }{(t^2-mathbf{x}^2)^Delta} $$






      complex-analysis




      complex-analysis






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 7 '18 at 13:13







      Craig Thone

















      asked Dec 7 '18 at 3:46









      Craig ThoneCraig Thone

      1727




      1727






















          0






          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',
          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%2f3029438%2fhow-to-calculate-contour-integration-of-multivalued-functions%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          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%2f3029438%2fhow-to-calculate-contour-integration-of-multivalued-functions%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

          Quarter-circle Tiles

          Image
          up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
          Read more

          build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

          Image
          0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
          Read more

          Mont Emei

          Image
          Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
          Read more