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The integral in question is: 

integrand<-function(y){
   exp(-sqrt(2*y + alpha^2)*abs(x))/ (pi^2 * y * ((besselJ(delta*sqrt(2*y), lambda))^2)
   + (besselY(delta*sqrt(2*y), lambda))^2)
}

integral<-function(x){integrate(integrand, lower=0, upper=Inf, subdivisions=20000)$value}

For some reason, when I try to run this (specifically for very small values of x), I get the error "The integral is probably divergent." I assumed that, since I was dividing by 'y', this was causing the error. However, if I change the equation around, ie, 

integrand<-function(y){
   (1/y)*(exp(-sqrt(2*y + alpha^2)*abs(x))/ (pi^2 * ((besselJ(delta*sqrt(2*y), lambda))^2)
   + (besselY(delta*sqrt(2*y), lambda))^2))
}

integral<-function(x){integrate(integrand, lower=0, upper=Inf, subdivisions=20000)$value}

Unless I'm mistaken (which I quite likely am) this should still be running the same equation. (Since multiplying by '1/y' should yield the same result as dividing by 'y'). However, running the integral this way, I don't get the "The integral is probably divergent" error. I'm very confused, and clearly I'm doing something terribly wrong here!

Edit: This was in fact not a duplicate of another question. The reason I was getting the error was due to a missing pair of brackets, not due to the tolerance as per the other question.

Chris Bridgett
  • 9
  • 5

1 Answers1

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Okay, so I solved it all by myself!

There was a pair of brackets missing around the Bessel functions' addition; the function should read: 

integrand<-function(y){
   exp(-sqrt(2*y + alpha^2)*abs(x))/ (pi^2 * y * (((besselJ(delta*sqrt(2*y), lambda))^2)
   + (besselY(delta*sqrt(2*y), lambda))^2))
}

integral<-function(x){
   integrate(integrand, lower=0, upper=Inf, subdivisions=20000)$value
}
Chris Bridgett
  • 9
  • 5