Gaussian integral over the rindler wedge

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I'm trying to evaluate analytically, if it's possible, the Gaussian integral over the Rindler wedge, meaning:



$int^infty_0dx_1int_-x_1^x_1dx_0e^-b(x_1+a_1)^2e^b(x_0+a_0)^2$



Where $a_1,a_0$ are just real constants.
As you can see this is an integral over the infinite cone with an angle that goes from $theta=-pi/4 $ to $theta=pi/4$, where $theta=0$ placed on the $x_1$ axis.



I tried to integrate first $x_0$ and I got Erfi functions but after I tried to integrate the second parameter Mathematica couldn't figure out the answer.



Is there a way to solve this analytically?










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    I'm trying to evaluate analytically, if it's possible, the Gaussian integral over the Rindler wedge, meaning:



    $int^infty_0dx_1int_-x_1^x_1dx_0e^-b(x_1+a_1)^2e^b(x_0+a_0)^2$



    Where $a_1,a_0$ are just real constants.
    As you can see this is an integral over the infinite cone with an angle that goes from $theta=-pi/4 $ to $theta=pi/4$, where $theta=0$ placed on the $x_1$ axis.



    I tried to integrate first $x_0$ and I got Erfi functions but after I tried to integrate the second parameter Mathematica couldn't figure out the answer.



    Is there a way to solve this analytically?










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I'm trying to evaluate analytically, if it's possible, the Gaussian integral over the Rindler wedge, meaning:



      $int^infty_0dx_1int_-x_1^x_1dx_0e^-b(x_1+a_1)^2e^b(x_0+a_0)^2$



      Where $a_1,a_0$ are just real constants.
      As you can see this is an integral over the infinite cone with an angle that goes from $theta=-pi/4 $ to $theta=pi/4$, where $theta=0$ placed on the $x_1$ axis.



      I tried to integrate first $x_0$ and I got Erfi functions but after I tried to integrate the second parameter Mathematica couldn't figure out the answer.



      Is there a way to solve this analytically?










      share|cite|improve this question













      I'm trying to evaluate analytically, if it's possible, the Gaussian integral over the Rindler wedge, meaning:



      $int^infty_0dx_1int_-x_1^x_1dx_0e^-b(x_1+a_1)^2e^b(x_0+a_0)^2$



      Where $a_1,a_0$ are just real constants.
      As you can see this is an integral over the infinite cone with an angle that goes from $theta=-pi/4 $ to $theta=pi/4$, where $theta=0$ placed on the $x_1$ axis.



      I tried to integrate first $x_0$ and I got Erfi functions but after I tried to integrate the second parameter Mathematica couldn't figure out the answer.



      Is there a way to solve this analytically?







      gaussian-integral






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      asked Sep 8 at 10:44









      Noam Chai

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