A suspicious result of an integral

I integrated this term in Mathematica:

$$\int_{-\infty}^{\infty} d\omega*\sin(s*\omega)*\frac{1}{e^{\beta*\hbar*\omega}-1}*\frac{\omega}{\omega^{2}+\gamma^{2}}$$

The code in Mathematica:

Integrate[Sin[s*ω]*(1/(Exp[β*ℏ*ω] -1))*ω/(ω^2 + γ^2), {ω, -Infinity, Infinity},
Assumptions -> {ℏ > 0, s >= 0, γ > 0, β > 0}]

The result is:

$\frac{-\pi}{2}*e^{-s*\gamma}$

Where is $\beta*\hbar$?

• The result is correct. The integral does not depend on either $\beta$ or $\hbar$, which is why $\beta\hbar$ does not appear in your answer. Commented Jun 10, 2014 at 14:00
• So we could set $\beta$=0 ? Commented Jun 10, 2014 at 14:03
• β>0 is in the Assumptions.. Commented Jun 10, 2014 at 14:17

we can see why this happens with a change of variable:

Integrate[bhw Sin[(bhw s)/bh]/((-1 + E^bhw) (bhw^2 + bh^2 gamma^2)),
{bhw, -Infinity, Infinity}, Assumptions -> {s >= 0, gamma > 0, bh > 0}]

(same result )

Now change the one instance of bh to a new parameter:

Integrate[bhw Sin[(bhw s)/bh1]/((-1 + E^bhw) (bhw^2 + bh^2 gamma^2)),
{bhw, -Infinity, Infinity}, Assumptions -> {s >= 0, gamma > 0, bh > 0,bh1>0}]

-(1/2) E^(-((bh s gamma )/bh1)) Pi

So you see with bh=bh1 they cancel out..

• If we take $\beta$ infinitely small, expand the exponential, and keep the lowest term then the result would be zero. However if we want to take the limit of $\beta ->0$ in the result we won't get zero.
– MOON
Commented Jun 10, 2014 at 15:01
• you can change your assumption to β!=0 and readily see there is a discontinuity (change of sign) at zero so that the limit does not exist. Commented Jun 10, 2014 at 15:23
• So I cannot expand the exponential. right?
– MOON
Commented Jun 10, 2014 at 15:36
• You ought to ask on math.stackexchange.com if you want to study why that method of taking the limit isn't working. Commented Jun 10, 2014 at 15:49

Another way to see the result is to do it by contour integration.

Start by expanding out the integrand.

(Sin[s omega] omega)/((Exp[beta hbar omega] - 1) (omega^2 + gamma^2)) // TrigToExp // Expand

(*
(I E^(-I omega s) omega)/(2 (-1 + E^(beta hbar omega)) (gamma^2 + omega^2))
- (I E^(I omega s) omega)/(2 (-1 + E^(beta hbar omega)) (gamma^2 + omega^2))
*)

For s > 0 the first term needs the contour to be closed towards omega = -I Infinity (i.e contributes -2 Pi I times the residue at omega = -I gamma), and the second term needs the contour to be closed towards omega = I Infinity (i.e contributes 2 Pi I times the residue at omega = I gamma).

This gives

-2 \[Pi] I Residue[(I E^(-I omega s) omega)/(2 (-1 + E^(beta hbar omega)) (gamma^2 + omega^2)), {omega, -I gamma}] +
2 \[Pi] I Residue[-((I E^(I omega s) omega)/(2 (-1 + E^(beta hbar omega)) (gamma^2 + omega^2))), {omega, I gamma}]

(*
(E^(-gamma s) \[Pi])/(2 (-1 + E^(I beta gamma hbar)))
- (E^(I beta gamma hbar - gamma s) \[Pi])/(2 (-1 + E^(I beta gamma hbar)))
*)

which simplifies thus

% // Simplify

(* -(1/2) E^(-gamma s) \[Pi] *)

The case s = 0 is not covered by the above derivation, but the integral is clearly 0 in this special case.

Edit (to correct an omission pointed out by Daniel Lichtblau in the comments below):

There are also poles where Exp[beta hbar omega] == 1, but these cancel in pairs as follows:

res[m_] =
SeriesCoefficient[(
Sin[s omega] omega)/((Exp[beta hbar omega] - 1) (omega^2 + gamma^2)), {omega, (2 I \[Pi] m)/(beta hbar), -1},
Assumptions -> m \[Element] Integers]

(*
(2 E^(-2 I m \[Pi]) m \[Pi] Sinh[(2 m \[Pi] s)/(beta hbar)])/(-beta^2 gamma^2 hbar^2 + 4 m^2 \[Pi]^2)
*)

Combine the corresponding residues from poles on the positive and negative imaginary axis, using opposite signs because the contour-completion at +I Infinity or -I Infinity goes round the poles in opposite directions.

res[m] - res[-m] // Simplify[#, Assumptions -> m \[Element] Integers] &

(* 0 *)

So these additional poles do not contribute to the integral.

• Aren't there other poles on the imaginary axis, where Exp[beta hbar omega] == 1? (Maybe those residues sum to zero?) Commented Jun 10, 2014 at 23:55
• Yes, you are right, I overlooked those poles in my haste to post an "answer" that seemed to explain everything. However, I got lucky, because the poles where Exp[beta hbar omega] == 1 cancel in pairs between the positive/negative imaginary axis. Commented Jun 11, 2014 at 13:10
• I had also gone down the contour path. You got further than I did-- I didn't do the work to figure out those "main" poles gave a viable result, let alone notice the cancellation amongst the rest. When I saw you had a correct value I just figured something like that must have happened. Commented Jun 11, 2014 at 13:38