Does this integral converge? $int frac{d^2x}{(2pi)^2} frac{d^2y}{(2pi)^2} [dots] exp [- frac{1}{4pi i}(x_1^2...












1












$begingroup$


Here I have a generalization of the Fresnel integral in two variables, and I'd like to check that it converges:



$$ frac{1}{2} int_{mathbb{R}^2 times mathbb{R}^2} frac{d^2x}{(2pi)^2} frac{d^2y}{(2pi)^2}
frac{big[2sinh big(frac{x_1 - x_2}{2}big)big]^2big[2sinh big(frac{y_1 - y_2}{2}big)big]^2}{prod_{i,j=1,2}big[2 cosh big(frac{x_i - y_j}{2}big)big]^2}
exp left[- frac{1}{4pi i}Big((x_1^2 - y_1^2)+(x_2^2- y_2^2)Big)right]$$



The domain of integration is that $(x_1, x_2)$ and $(y_1, y_2)$ vary all over $mathbb{R}^2$. This is from a physics computation.



Or prove that it diverges. This integral might give two different answers depending on which subspace we integrate first:




  • ${x_1 = x_2} cap {y_1 = y_2}$ so that $sinh (x_1 - x_2) = sinh 0 = 0$.


  • ${ x_1 = y_1} cap {x_2 = y_2}$ so that $e^{x_1^2 - y_1^2} = e^{x_2^2 - y_2^2} = e^0 = 1$


  • ${ x_1 = y_2} cap {x_2 = y_1}$ so that $e^{x_1^2 - y_2^2} = e^{x_2^2 - y_1^2}= e^0 = 1$.



In the first case - without integrating over the remaining subspace $mathbb{R}^4 / big(mathbb{R}(1,-1) oplus mathbb{R}(1,-1)big) simeq mathbb{R}^2$ - we'd have that $int = 0$. My notes have $int = frac{1}{16}$ and I wonder what the other two iterated integrals have us.





This looks like it could be realted to the Fresnel integral.



$$ int_0^infty sin x^2 , dx = int_0^infty cos x^2 , dx = frac{sqrt{2pi}}{4} $$










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$endgroup$

















    1












    $begingroup$


    Here I have a generalization of the Fresnel integral in two variables, and I'd like to check that it converges:



    $$ frac{1}{2} int_{mathbb{R}^2 times mathbb{R}^2} frac{d^2x}{(2pi)^2} frac{d^2y}{(2pi)^2}
    frac{big[2sinh big(frac{x_1 - x_2}{2}big)big]^2big[2sinh big(frac{y_1 - y_2}{2}big)big]^2}{prod_{i,j=1,2}big[2 cosh big(frac{x_i - y_j}{2}big)big]^2}
    exp left[- frac{1}{4pi i}Big((x_1^2 - y_1^2)+(x_2^2- y_2^2)Big)right]$$



    The domain of integration is that $(x_1, x_2)$ and $(y_1, y_2)$ vary all over $mathbb{R}^2$. This is from a physics computation.



    Or prove that it diverges. This integral might give two different answers depending on which subspace we integrate first:




    • ${x_1 = x_2} cap {y_1 = y_2}$ so that $sinh (x_1 - x_2) = sinh 0 = 0$.


    • ${ x_1 = y_1} cap {x_2 = y_2}$ so that $e^{x_1^2 - y_1^2} = e^{x_2^2 - y_2^2} = e^0 = 1$


    • ${ x_1 = y_2} cap {x_2 = y_1}$ so that $e^{x_1^2 - y_2^2} = e^{x_2^2 - y_1^2}= e^0 = 1$.



    In the first case - without integrating over the remaining subspace $mathbb{R}^4 / big(mathbb{R}(1,-1) oplus mathbb{R}(1,-1)big) simeq mathbb{R}^2$ - we'd have that $int = 0$. My notes have $int = frac{1}{16}$ and I wonder what the other two iterated integrals have us.





    This looks like it could be realted to the Fresnel integral.



    $$ int_0^infty sin x^2 , dx = int_0^infty cos x^2 , dx = frac{sqrt{2pi}}{4} $$










    share|cite|improve this question











    $endgroup$















      1












      1








      1


      1



      $begingroup$


      Here I have a generalization of the Fresnel integral in two variables, and I'd like to check that it converges:



      $$ frac{1}{2} int_{mathbb{R}^2 times mathbb{R}^2} frac{d^2x}{(2pi)^2} frac{d^2y}{(2pi)^2}
      frac{big[2sinh big(frac{x_1 - x_2}{2}big)big]^2big[2sinh big(frac{y_1 - y_2}{2}big)big]^2}{prod_{i,j=1,2}big[2 cosh big(frac{x_i - y_j}{2}big)big]^2}
      exp left[- frac{1}{4pi i}Big((x_1^2 - y_1^2)+(x_2^2- y_2^2)Big)right]$$



      The domain of integration is that $(x_1, x_2)$ and $(y_1, y_2)$ vary all over $mathbb{R}^2$. This is from a physics computation.



      Or prove that it diverges. This integral might give two different answers depending on which subspace we integrate first:




      • ${x_1 = x_2} cap {y_1 = y_2}$ so that $sinh (x_1 - x_2) = sinh 0 = 0$.


      • ${ x_1 = y_1} cap {x_2 = y_2}$ so that $e^{x_1^2 - y_1^2} = e^{x_2^2 - y_2^2} = e^0 = 1$


      • ${ x_1 = y_2} cap {x_2 = y_1}$ so that $e^{x_1^2 - y_2^2} = e^{x_2^2 - y_1^2}= e^0 = 1$.



      In the first case - without integrating over the remaining subspace $mathbb{R}^4 / big(mathbb{R}(1,-1) oplus mathbb{R}(1,-1)big) simeq mathbb{R}^2$ - we'd have that $int = 0$. My notes have $int = frac{1}{16}$ and I wonder what the other two iterated integrals have us.





      This looks like it could be realted to the Fresnel integral.



      $$ int_0^infty sin x^2 , dx = int_0^infty cos x^2 , dx = frac{sqrt{2pi}}{4} $$










      share|cite|improve this question











      $endgroup$




      Here I have a generalization of the Fresnel integral in two variables, and I'd like to check that it converges:



      $$ frac{1}{2} int_{mathbb{R}^2 times mathbb{R}^2} frac{d^2x}{(2pi)^2} frac{d^2y}{(2pi)^2}
      frac{big[2sinh big(frac{x_1 - x_2}{2}big)big]^2big[2sinh big(frac{y_1 - y_2}{2}big)big]^2}{prod_{i,j=1,2}big[2 cosh big(frac{x_i - y_j}{2}big)big]^2}
      exp left[- frac{1}{4pi i}Big((x_1^2 - y_1^2)+(x_2^2- y_2^2)Big)right]$$



      The domain of integration is that $(x_1, x_2)$ and $(y_1, y_2)$ vary all over $mathbb{R}^2$. This is from a physics computation.



      Or prove that it diverges. This integral might give two different answers depending on which subspace we integrate first:




      • ${x_1 = x_2} cap {y_1 = y_2}$ so that $sinh (x_1 - x_2) = sinh 0 = 0$.


      • ${ x_1 = y_1} cap {x_2 = y_2}$ so that $e^{x_1^2 - y_1^2} = e^{x_2^2 - y_2^2} = e^0 = 1$


      • ${ x_1 = y_2} cap {x_2 = y_1}$ so that $e^{x_1^2 - y_2^2} = e^{x_2^2 - y_1^2}= e^0 = 1$.



      In the first case - without integrating over the remaining subspace $mathbb{R}^4 / big(mathbb{R}(1,-1) oplus mathbb{R}(1,-1)big) simeq mathbb{R}^2$ - we'd have that $int = 0$. My notes have $int = frac{1}{16}$ and I wonder what the other two iterated integrals have us.





      This looks like it could be realted to the Fresnel integral.



      $$ int_0^infty sin x^2 , dx = int_0^infty cos x^2 , dx = frac{sqrt{2pi}}{4} $$







      improper-integrals multiple-integral






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      edited Dec 16 '18 at 21:46







      cactus314

















      asked Dec 16 '18 at 21:01









      cactus314cactus314

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