Skip to main content
Mathematica

Return to Question

edited title
Link
David G. Stork
David G. Stork
  • 42.3k
  • 3
  • 37
  • 106

Interesting discrepencies between integrate functions - what is happening?

added 1714 characters in body
Source Link
Adam
Adam
  • 63
  • 3

So, NIntegrate and Integrate disagree by exactly a factor of 1/2 and moreoever my exact calculation agrees exactly with NIntegrate. It seems like Integrate is getting something wrong? Could this be to do with taking the principal value? Another odd thing is that when you split the principal part integral up into the sum of two pricipalprincipal part integrals Integrate doesn't give you the same answer, so the principal part integration is apparently not linear? Can anybody explain what is happening here?

Thanks!

Later amendment:

Yet further odd behaviour is found if you Simplify the integrand. By fiddling around with the integrand one can pin down discrepencies in the results of Integrate based on how the integrand's denominator is expressed. Specifically

Integrate[(
 2 c (c R wk Cos[(R wk)/c] + (-c^2 + R^2 wk^2) Sin[(R wk)/c]))/(
 R^3 wk (w0 - wk) (w0 + wk)), {wk, 0, Infinity}, 
 PrincipalValue -> True]

gives exactly 2/3 the result of

Integrate[(
 2 c (c R wk Cos[(R wk)/c] + (-c^2 + R^2 wk^2) Sin[(R wk)/c]))/(
 R^3 wk (w0^2 - wk^2)), {wk, 0, Infinity}, 
 PrincipalValue -> True]

yet the only difference is that I've multiplied out the brackets in the denominator of the first integrand to get the denominator in the second integrand, that is, I used

(w0 - wk) (w0 + wk) = w0^2-wk^2.

There is clearly something wrong with Integrate for this to be happening?! If you Integrate the expression which results from taking the difference of the two integrands above then you get the message

Infinity::indet: Indeterminate expression 0. \[Infinity] encountered.

along with the answer {0.}.

If you take the difference of the integrands and then Simplify the result you get 0 obviously and then the integral evaluates to 0 identically. But evaluating the integrals separately and then taking the difference gives you a non-zero result because mathematica seems to think that the two integrals differ by a factor of 2/3, even though the integrands are identically equal! It seems that Integrate and PrincipalValue are somewhat unreliable for these problems? Does anybody know if this is to do with the PrincipalValue?

Thanks.

So, NIntegrate and Integrate disagree by exactly a factor of 1/2 and moreoever my exact calculation agrees exactly with NIntegrate. It seems like Integrate is getting something wrong? Could this be to do with taking the principal value? Another odd thing is that when you split the principal part integral up into the sum of two pricipal part integrals Integrate doesn't give you the same answer, so the principal part integration is apparently not linear? Can anybody explain what is happening here?

Thanks!

So, NIntegrate and Integrate disagree by exactly a factor of 1/2 and moreoever my exact calculation agrees exactly with NIntegrate. It seems like Integrate is getting something wrong? Could this be to do with taking the principal value? Another odd thing is that when you split the principal part integral up into the sum of two principal part integrals Integrate doesn't give you the same answer, so the principal part integration is apparently not linear? Can anybody explain what is happening here?

Thanks!

Later amendment:

Yet further odd behaviour is found if you Simplify the integrand. By fiddling around with the integrand one can pin down discrepencies in the results of Integrate based on how the integrand's denominator is expressed. Specifically

Integrate[(
 2 c (c R wk Cos[(R wk)/c] + (-c^2 + R^2 wk^2) Sin[(R wk)/c]))/(
 R^3 wk (w0 - wk) (w0 + wk)), {wk, 0, Infinity}, 
 PrincipalValue -> True]

gives exactly 2/3 the result of

Integrate[(
 2 c (c R wk Cos[(R wk)/c] + (-c^2 + R^2 wk^2) Sin[(R wk)/c]))/(
 R^3 wk (w0^2 - wk^2)), {wk, 0, Infinity}, 
 PrincipalValue -> True]

yet the only difference is that I've multiplied out the brackets in the denominator of the first integrand to get the denominator in the second integrand, that is, I used

(w0 - wk) (w0 + wk) = w0^2-wk^2.

There is clearly something wrong with Integrate for this to be happening?! If you Integrate the expression which results from taking the difference of the two integrands above then you get the message

Infinity::indet: Indeterminate expression 0. \[Infinity] encountered.

along with the answer {0.}.

If you take the difference of the integrands and then Simplify the result you get 0 obviously and then the integral evaluates to 0 identically. But evaluating the integrals separately and then taking the difference gives you a non-zero result because mathematica seems to think that the two integrals differ by a factor of 2/3, even though the integrands are identically equal! It seems that Integrate and PrincipalValue are somewhat unreliable for these problems? Does anybody know if this is to do with the PrincipalValue?

Thanks.

Source Link
Adam
Adam
  • 63
  • 3

Interesting discrepencies between integrate functions - what is happening?

I am currently trying to compute some integrals for a physics problem. I have a number of real parameters which are

w0 = 10^(10);
d = 3.18*10^(-26);
ra = 1.33*10^(-7);
eps = 8.85*10^{-12};
hbar = 1.05*10^{-34};
c = 3*10^8;
R = 15 ra;.

I also define a function

g[w0_] := w0^3 d^2/(3 Pi eps hbar c^3).

Now, I tried to perform the following integration numerically with NIntegrate

Int=(3 g[w0]/(4 Pi w0)) NIntegrate[(wk/(w0 - wk)) (Sin[wk R/c]/(wk R/c)+  
      Cos[wk R/c]/(wk R/c)^2 - 
      Sin[wk R/c]/(wk R/c)^3) - (wk/(w0 + wk)) (Sin[
        wk R/c]/(wk R/c) + Cos[wk R/c]/(wk R/c)^2 - 
      Sin[wk R/c]/(wk R/c)^3), {wk, 0, Infinity}, MaxRecursion -> 100]

I'm told the numerical integration is converging too slowly, but I still get an answer which is

Int=-24.1155.

At this point I don't really know what to make of this answer, so I try using Integrate instead of NIntegrate. I perform the integral

Int'=(3 g[w0]/(4 Pi w0)) Integrate[(wk/(w0 - wk)) (Sin[wk R/c]/(wk R/c) + 
      Cos[wk R/c]/(wk R/c)^2 - 
      Sin[wk R/c]/(wk R/c)^3) - (wk/(w0 + wk)) (Sin[
        wk R/c]/(wk R/c) + Cos[wk R/c]/(wk R/c)^2 - 
      Sin[wk R/c]/(wk R/c)^3), {wk, 0, Infinity}, 
  PrincipalValue -> True]

which gives me the answer

Int'=-48.231 + 0.00320736 I.

The first interesting thing is that

Re[Int']=2Int,

the numerical integral is giving me exactly half the answer of Integrate in the real part. Things get even more interesting by noting that one can actually compute this integral analytically with residues, which gives

Int=-(d^2 w0^3/(4 Pi hbar c^3 eps)) (Cos[w0 R/c]/(w0 R/c) - Sin[w0 R/c]/(w0 R/c)^2
     +(1 - Cos[w0 R/c])/(w0 R/c)^3),

which for the parameters I give at the start of the post gives me the answer

Int = -24.1155.

So, NIntegrate and Integrate disagree by exactly a factor of 1/2 and moreoever my exact calculation agrees exactly with NIntegrate. It seems like Integrate is getting something wrong? Could this be to do with taking the principal value? Another odd thing is that when you split the principal part integral up into the sum of two pricipal part integrals Integrate doesn't give you the same answer, so the principal part integration is apparently not linear? Can anybody explain what is happening here?

Thanks!