Highly oscillating integral [closed]

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I am trying to integrate a highly oscillatory integral. the integrand is a function of 4 variables: 



f1, f2, x2, y


the integrand is



int= (768*E^(8*I*y)*f1*(1 - f1 - f2)*f2*Pi^6*x2*If[1 >= f1 + f2, 1, 0]*
 ((-4*x2*(319 + 239*x2^2) + (533 + 2388*x2^2 + 370*x2^4)*Cos[y] - 
 4*x2*(113 + 178*x2^2)*Cos[2*y] - (139 - 138*x2^2 - 50*x2^4)*Cos[3*y] - 
 4*x2*(-13 + 3*x2^2)*Cos[4*y] - (-29 + 6*x2^2)*Cos[5*y] - 4*x2*Cos[6*y] - 
 3*Cos[7*y])*Log[1 + x2^2 - 2*x2*Cos[y]] + 
 x2*(2*(-101 - 31*x2^2 + 113*Cos[2*y] - 13*Cos[4*y] + Cos[6*y] + 420*Log[x2]) + 
 x2*(154*Cos[y] + 36*x2*Cos[2*y] - 161*Cos[3*y] + 26*x2*Cos[4*y] + 7*Cos[5*y] - 
 20*((126 + 37*x2^2)*Cos[y] + x2*(-74 - 52*Cos[2*y] + 5*x2*Cos[3*y]))*Log[x2]) - 
 (1/x2)*(12*(35 + 120*x2^2 + 18*x2^4 - 20*x2*(7 + 6*x2^2)*Cos[y] + 
 2*x2^2*(45 + 8*x2^2)*Cos[2*y] - 20*x2^3*Cos[3*y] + x2^4*Cos[4*y])*Csc[y]*
 (y*(Log[1 - x2/E^(I*y)] + Log[1 - E^(I*y)*x2]) + I*PolyLog[2, 1 - x2/E^(I*y)] - 
 I*PolyLog[2, 1 - E^(I*y)*x2]))))*Sin[y])/
 ((-1 + E^(2*I*y))^8*(f1 - f1^2 + f2*x2^2 - f2^2*x2^2 - 2*f1*f2*x2*Cos[y])^2)


the integral i want to perform is



NIntegrate[int, f1, 0, 1, f2, 0, 1, x2, 0, [Infinity], y, 
 0, [Pi]]


The oscillations occur for the variable y near 0 and infinity. If you plot int (setting f1,f2,x2 to some random variables, they dont matter), you will see that the function is highly oscillating in these ranges. for instance



Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 3, [Pi]]
Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 0, .1]


enter image description hereenter image description here




numerical-methods oscillatory-integral




share|cite|improve this question











closed as off-topic by quid♦ Jul 31 at 15:03



  • This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I would ask you write the integrand in MathJax, but it is nightmarish. Can you explain why you are interested in calculating this seemingly arbitrary integral?
    – packetpacket
    Jul 30 at 19:36










  • this integral (as well as 14 others like it) arises when performing an integral over a certain 4-point correlation function of a conformal field theory. the integral was originally 12-dimensional, but i managed to simplify to just these 4 variables. i actually expect that the sum of all these integrals should be a relatively simple number, just a rational number times a power of pi. i was hoping to be able to check this conjecture numerically.
    – esches
    Jul 30 at 19:49










  • ok ill do that, thanks!
    – esches
    Jul 30 at 20:23










  • Given that the above is essentially unreadable by humans, could you at least abstract out the integral into a few pieces, with defined functions? For example, group stuff like $p(x)cos(kx)$ where $p$ is a polynomial, the polylogs and logs. The constant coefficients involved seem irrelevant, except perhaps for the logs and polylogs.
    – Alex R.
    Jul 31 at 0:14











  • I'm voting to close this question because it was asked and answered on Mathematica : mathematica.stackexchange.com/questions/179252/…
    – quid♦
    Jul 31 at 15:03














up vote
1
down vote

favorite












I am trying to integrate a highly oscillatory integral. the integrand is a function of 4 variables: 



f1, f2, x2, y


the integrand is



int= (768*E^(8*I*y)*f1*(1 - f1 - f2)*f2*Pi^6*x2*If[1 >= f1 + f2, 1, 0]*
 ((-4*x2*(319 + 239*x2^2) + (533 + 2388*x2^2 + 370*x2^4)*Cos[y] - 
 4*x2*(113 + 178*x2^2)*Cos[2*y] - (139 - 138*x2^2 - 50*x2^4)*Cos[3*y] - 
 4*x2*(-13 + 3*x2^2)*Cos[4*y] - (-29 + 6*x2^2)*Cos[5*y] - 4*x2*Cos[6*y] - 
 3*Cos[7*y])*Log[1 + x2^2 - 2*x2*Cos[y]] + 
 x2*(2*(-101 - 31*x2^2 + 113*Cos[2*y] - 13*Cos[4*y] + Cos[6*y] + 420*Log[x2]) + 
 x2*(154*Cos[y] + 36*x2*Cos[2*y] - 161*Cos[3*y] + 26*x2*Cos[4*y] + 7*Cos[5*y] - 
 20*((126 + 37*x2^2)*Cos[y] + x2*(-74 - 52*Cos[2*y] + 5*x2*Cos[3*y]))*Log[x2]) - 
 (1/x2)*(12*(35 + 120*x2^2 + 18*x2^4 - 20*x2*(7 + 6*x2^2)*Cos[y] + 
 2*x2^2*(45 + 8*x2^2)*Cos[2*y] - 20*x2^3*Cos[3*y] + x2^4*Cos[4*y])*Csc[y]*
 (y*(Log[1 - x2/E^(I*y)] + Log[1 - E^(I*y)*x2]) + I*PolyLog[2, 1 - x2/E^(I*y)] - 
 I*PolyLog[2, 1 - E^(I*y)*x2]))))*Sin[y])/
 ((-1 + E^(2*I*y))^8*(f1 - f1^2 + f2*x2^2 - f2^2*x2^2 - 2*f1*f2*x2*Cos[y])^2)


the integral i want to perform is



NIntegrate[int, f1, 0, 1, f2, 0, 1, x2, 0, [Infinity], y, 
 0, [Pi]]


The oscillations occur for the variable y near 0 and infinity. If you plot int (setting f1,f2,x2 to some random variables, they dont matter), you will see that the function is highly oscillating in these ranges. for instance



Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 3, [Pi]]
Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 0, .1]


enter image description hereenter image description here




numerical-methods oscillatory-integral




share|cite|improve this question











closed as off-topic by quid♦ Jul 31 at 15:03



  • This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I would ask you write the integrand in MathJax, but it is nightmarish. Can you explain why you are interested in calculating this seemingly arbitrary integral?
    – packetpacket
    Jul 30 at 19:36










  • this integral (as well as 14 others like it) arises when performing an integral over a certain 4-point correlation function of a conformal field theory. the integral was originally 12-dimensional, but i managed to simplify to just these 4 variables. i actually expect that the sum of all these integrals should be a relatively simple number, just a rational number times a power of pi. i was hoping to be able to check this conjecture numerically.
    – esches
    Jul 30 at 19:49










  • ok ill do that, thanks!
    – esches
    Jul 30 at 20:23










  • Given that the above is essentially unreadable by humans, could you at least abstract out the integral into a few pieces, with defined functions? For example, group stuff like $p(x)cos(kx)$ where $p$ is a polynomial, the polylogs and logs. The constant coefficients involved seem irrelevant, except perhaps for the logs and polylogs.
    – Alex R.
    Jul 31 at 0:14











  • I'm voting to close this question because it was asked and answered on Mathematica : mathematica.stackexchange.com/questions/179252/…
    – quid♦
    Jul 31 at 15:03












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I am trying to integrate a highly oscillatory integral. the integrand is a function of 4 variables: 



f1, f2, x2, y


the integrand is



int= (768*E^(8*I*y)*f1*(1 - f1 - f2)*f2*Pi^6*x2*If[1 >= f1 + f2, 1, 0]*
 ((-4*x2*(319 + 239*x2^2) + (533 + 2388*x2^2 + 370*x2^4)*Cos[y] - 
 4*x2*(113 + 178*x2^2)*Cos[2*y] - (139 - 138*x2^2 - 50*x2^4)*Cos[3*y] - 
 4*x2*(-13 + 3*x2^2)*Cos[4*y] - (-29 + 6*x2^2)*Cos[5*y] - 4*x2*Cos[6*y] - 
 3*Cos[7*y])*Log[1 + x2^2 - 2*x2*Cos[y]] + 
 x2*(2*(-101 - 31*x2^2 + 113*Cos[2*y] - 13*Cos[4*y] + Cos[6*y] + 420*Log[x2]) + 
 x2*(154*Cos[y] + 36*x2*Cos[2*y] - 161*Cos[3*y] + 26*x2*Cos[4*y] + 7*Cos[5*y] - 
 20*((126 + 37*x2^2)*Cos[y] + x2*(-74 - 52*Cos[2*y] + 5*x2*Cos[3*y]))*Log[x2]) - 
 (1/x2)*(12*(35 + 120*x2^2 + 18*x2^4 - 20*x2*(7 + 6*x2^2)*Cos[y] + 
 2*x2^2*(45 + 8*x2^2)*Cos[2*y] - 20*x2^3*Cos[3*y] + x2^4*Cos[4*y])*Csc[y]*
 (y*(Log[1 - x2/E^(I*y)] + Log[1 - E^(I*y)*x2]) + I*PolyLog[2, 1 - x2/E^(I*y)] - 
 I*PolyLog[2, 1 - E^(I*y)*x2]))))*Sin[y])/
 ((-1 + E^(2*I*y))^8*(f1 - f1^2 + f2*x2^2 - f2^2*x2^2 - 2*f1*f2*x2*Cos[y])^2)


the integral i want to perform is



NIntegrate[int, f1, 0, 1, f2, 0, 1, x2, 0, [Infinity], y, 
 0, [Pi]]


The oscillations occur for the variable y near 0 and infinity. If you plot int (setting f1,f2,x2 to some random variables, they dont matter), you will see that the function is highly oscillating in these ranges. for instance



Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 3, [Pi]]
Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 0, .1]


enter image description hereenter image description here




numerical-methods oscillatory-integral




share|cite|improve this question











I am trying to integrate a highly oscillatory integral. the integrand is a function of 4 variables: 



f1, f2, x2, y


the integrand is



int= (768*E^(8*I*y)*f1*(1 - f1 - f2)*f2*Pi^6*x2*If[1 >= f1 + f2, 1, 0]*
 ((-4*x2*(319 + 239*x2^2) + (533 + 2388*x2^2 + 370*x2^4)*Cos[y] - 
 4*x2*(113 + 178*x2^2)*Cos[2*y] - (139 - 138*x2^2 - 50*x2^4)*Cos[3*y] - 
 4*x2*(-13 + 3*x2^2)*Cos[4*y] - (-29 + 6*x2^2)*Cos[5*y] - 4*x2*Cos[6*y] - 
 3*Cos[7*y])*Log[1 + x2^2 - 2*x2*Cos[y]] + 
 x2*(2*(-101 - 31*x2^2 + 113*Cos[2*y] - 13*Cos[4*y] + Cos[6*y] + 420*Log[x2]) + 
 x2*(154*Cos[y] + 36*x2*Cos[2*y] - 161*Cos[3*y] + 26*x2*Cos[4*y] + 7*Cos[5*y] - 
 20*((126 + 37*x2^2)*Cos[y] + x2*(-74 - 52*Cos[2*y] + 5*x2*Cos[3*y]))*Log[x2]) - 
 (1/x2)*(12*(35 + 120*x2^2 + 18*x2^4 - 20*x2*(7 + 6*x2^2)*Cos[y] + 
 2*x2^2*(45 + 8*x2^2)*Cos[2*y] - 20*x2^3*Cos[3*y] + x2^4*Cos[4*y])*Csc[y]*
 (y*(Log[1 - x2/E^(I*y)] + Log[1 - E^(I*y)*x2]) + I*PolyLog[2, 1 - x2/E^(I*y)] - 
 I*PolyLog[2, 1 - E^(I*y)*x2]))))*Sin[y])/
 ((-1 + E^(2*I*y))^8*(f1 - f1^2 + f2*x2^2 - f2^2*x2^2 - 2*f1*f2*x2*Cos[y])^2)


the integral i want to perform is



NIntegrate[int, f1, 0, 1, f2, 0, 1, x2, 0, [Infinity], y, 
 0, [Pi]]


The oscillations occur for the variable y near 0 and infinity. If you plot int (setting f1,f2,x2 to some random variables, they dont matter), you will see that the function is highly oscillating in these ranges. for instance



Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 3, [Pi]]
Plot[int /. f1 -> 1/2 /. f2 -> 1/3 /. x2 -> 1, y, 0, .1]


enter image description hereenter image description here





numerical-methods oscillatory-integral





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 19:27









esches

161




161




closed as off-topic by quid♦ Jul 31 at 15:03



  • This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by quid♦ Jul 31 at 15:03



  • This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.











  • I would ask you write the integrand in MathJax, but it is nightmarish. Can you explain why you are interested in calculating this seemingly arbitrary integral?
    – packetpacket
    Jul 30 at 19:36










  • this integral (as well as 14 others like it) arises when performing an integral over a certain 4-point correlation function of a conformal field theory. the integral was originally 12-dimensional, but i managed to simplify to just these 4 variables. i actually expect that the sum of all these integrals should be a relatively simple number, just a rational number times a power of pi. i was hoping to be able to check this conjecture numerically.
    – esches
    Jul 30 at 19:49










  • ok ill do that, thanks!
    – esches
    Jul 30 at 20:23










  • Given that the above is essentially unreadable by humans, could you at least abstract out the integral into a few pieces, with defined functions? For example, group stuff like $p(x)cos(kx)$ where $p$ is a polynomial, the polylogs and logs. The constant coefficients involved seem irrelevant, except perhaps for the logs and polylogs.
    – Alex R.
    Jul 31 at 0:14











  • I'm voting to close this question because it was asked and answered on Mathematica : mathematica.stackexchange.com/questions/179252/…
    – quid♦
    Jul 31 at 15:03
















  • I would ask you write the integrand in MathJax, but it is nightmarish. Can you explain why you are interested in calculating this seemingly arbitrary integral?
    – packetpacket
    Jul 30 at 19:36










  • this integral (as well as 14 others like it) arises when performing an integral over a certain 4-point correlation function of a conformal field theory. the integral was originally 12-dimensional, but i managed to simplify to just these 4 variables. i actually expect that the sum of all these integrals should be a relatively simple number, just a rational number times a power of pi. i was hoping to be able to check this conjecture numerically.
    – esches
    Jul 30 at 19:49










  • ok ill do that, thanks!
    – esches
    Jul 30 at 20:23










  • Given that the above is essentially unreadable by humans, could you at least abstract out the integral into a few pieces, with defined functions? For example, group stuff like $p(x)cos(kx)$ where $p$ is a polynomial, the polylogs and logs. The constant coefficients involved seem irrelevant, except perhaps for the logs and polylogs.
    – Alex R.
    Jul 31 at 0:14











  • I'm voting to close this question because it was asked and answered on Mathematica : mathematica.stackexchange.com/questions/179252/…
    – quid♦
    Jul 31 at 15:03















I would ask you write the integrand in MathJax, but it is nightmarish. Can you explain why you are interested in calculating this seemingly arbitrary integral?
– packetpacket
Jul 30 at 19:36




I would ask you write the integrand in MathJax, but it is nightmarish. Can you explain why you are interested in calculating this seemingly arbitrary integral?
– packetpacket
Jul 30 at 19:36












this integral (as well as 14 others like it) arises when performing an integral over a certain 4-point correlation function of a conformal field theory. the integral was originally 12-dimensional, but i managed to simplify to just these 4 variables. i actually expect that the sum of all these integrals should be a relatively simple number, just a rational number times a power of pi. i was hoping to be able to check this conjecture numerically.
– esches
Jul 30 at 19:49




this integral (as well as 14 others like it) arises when performing an integral over a certain 4-point correlation function of a conformal field theory. the integral was originally 12-dimensional, but i managed to simplify to just these 4 variables. i actually expect that the sum of all these integrals should be a relatively simple number, just a rational number times a power of pi. i was hoping to be able to check this conjecture numerically.
– esches
Jul 30 at 19:49












ok ill do that, thanks!
– esches
Jul 30 at 20:23




ok ill do that, thanks!
– esches
Jul 30 at 20:23












Given that the above is essentially unreadable by humans, could you at least abstract out the integral into a few pieces, with defined functions? For example, group stuff like $p(x)cos(kx)$ where $p$ is a polynomial, the polylogs and logs. The constant coefficients involved seem irrelevant, except perhaps for the logs and polylogs.
– Alex R.
Jul 31 at 0:14





Given that the above is essentially unreadable by humans, could you at least abstract out the integral into a few pieces, with defined functions? For example, group stuff like $p(x)cos(kx)$ where $p$ is a polynomial, the polylogs and logs. The constant coefficients involved seem irrelevant, except perhaps for the logs and polylogs.
– Alex R.
Jul 31 at 0:14













I'm voting to close this question because it was asked and answered on Mathematica : mathematica.stackexchange.com/questions/179252/…
– quid♦
Jul 31 at 15:03




I'm voting to close this question because it was asked and answered on Mathematica : mathematica.stackexchange.com/questions/179252/…
– quid♦
Jul 31 at 15:03















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