Integration over Dirac Deltas

Questions
1

The following integration returns back the input, but the integral can actually be evaluated by hand. What is wrong with this -- is it Mathematica 9 that does this?

Integrate[DiracDelta[-k1-q1] DiracDelta[k1-pm] DiracDelta[pm-(-q1)] *F1[k1]*w[pm,m]*Conjugate[F1[q1]],{pm,-Infinity,Infinity}]
2

Is it possible you haven’t specified m, q1 and k1?

3

Hello,

There is no problem solving this integral:

Integrate[DiracDelta[k1 - pm] DiracDelta[-k1 + q1] w[pm, m],{pm,-Infinity,Infinity}]

which gives

ConditionalExpression[DiracDelta[-k1 + q1] w[k1, m], k1 \[Element] Reals]

Just the added third DiracDelta seems to be the problem, but I don’t know why.

4

Note that the original product of 3 delta functions reduces to DiracDelta[0] before integration. (Recall delta(x1-x0)*delta(x2-x0)=delta(x1-x2), etc.) Thus the problem must be incorrectly formulated.

5

Dear Dr. Blinder,

Thanks for your feedback. The original formulation seems to be consistent to my eyes:
2 particles in CM frame with momenta q1 and -q1 scattering like billiard balls to final momenta k1 and -k1 through a simple free Hamiltonian with momentum “pm”

6

Naively speaking, DiracDelta[-k1-q1] would factor out of the integral and cause a singularity (if the DiracDelta combination were not evaluated inside the integral). Perhaps that’s why?

7

It looks like neither Mathematica nor Maple can correctly use things like
\delta(0) \delta(x) = \delta^2(0) etc.

8

I’m speaking a bit beyond my knowledge, but…

Integrate essentially uses the Risch algorithm, which I understand works for functions, but not distributions. DiracDelta and other things that aren’t actually functions can be problematic. Integrate seems to have some functionality though to handle distributions as best as it can. If you’d like, you can forward the example to the technical support team (support@wolfram.com) and suggest that Integrate be extended to handle . It’s proably not a bug, but maybe Integrate could be improved.

This is a common surprise for physicists using computer algebra. Another is that the results given don’t always seem to be very cooperative (esp with orthogonal functions) :

Integrate[Cos[m x] Cos[n x], {x, -Pi, Pi}]

gives:

(2 m Cos[n \[Pi]] Sin[m \[Pi]] - 2 n Cos[m \[Pi]] Sin[n \[Pi]])/(m^2 - n^2)

Where you’d might expect Pi*DiracDelta[m-n].