D: Dirac Delta Distributions

You have probably been introduced to the Dirac delta distribution (com- monly mis-named the Dirac delta function) in your ODE course while dis- cussing the Laplace transform technique, but probably not in the way we discuss it below. From the forced string problem

ρutt = T uxx + f0(x)(ρ = density/length, T = tension) in the equilibrium case we write d2u = −f(x) (with f := f (x)/T ) dx2 0 and consider f a distributed force per unit length. We can imagine a point force too (at x = a; see figure 1): d2u = 0 0 < x < l , x 6= a ∈ (0, l) . dx2 Can we represent such a point force within the differential equation? Approximate the point force by a sequence of functions f(x) that have an ever shrinking support 1. One example would be  1 − + a < x < a +   2 f(x) =  0 otherwise R We see that f(x) dx = 1 for any  > 0, over any interval containing support (a − , a + ), and for fixed x 6= a, f(x) → 0 as  → 0. Here Z x  0 0 1 x > a f(x ) dx → as  → 0 . −∞ 0 x < a R Also, for any φ(x) that is continuous in an interval about a, f(x)φ(x) dx → φ(a) as  → 0; that is, all the mass of f, hence f(x)φ(x) becomes concen- trated at x = a. If in an informal sense we write “lim→0f(x)” = δ(x−a), then we expect this limit to satisfy

1Support of a function is the set of values x where the function is nonzero.

1 Figure 1: Steady point force applied to an inextensible string.

R i. I δ(x) dx = 1, I is any interval about x = a ii. δ(x − a) = 0 for x 6= a

R x 0 0 iii. −∞ δ(x − a) dx = H(x − a) ,H(·) being the Heaviside function R iv. for smooth φ in interval I, I δ(x − a)φ(x) dx = φ(a). So δ is the Dirac delta distribution if it is defined on R and satisfies (i)-(iv). For the 2D version of the Dirac delta distribution (readily generalizble to higher dimensions), we write δ(x − x0), x = (x, y), x0 = (ξ, η), so we can also write δ(x − ξ, y − η), which we can interpret as a product of 1D delta distributions: δ(x − ξ, y − η) = δ(x − ξ)δ(y − η). Of course, to make analytic sense of all these claims in this appendix would require us to go into distribution theory, which is beyond the level of this course. That said, the formal notion of the Dirac delta distribution is a very handy concept in an applied sense.

Remark: We can√ expect that the fundamental solution to the heat equation, −x2/4 S(x, ) = e / 4π would act like the f above; that is, formally S(x, ) → δ(x) as  → 0. So the family of Gaussian (“bell” curves) converge to a unit point force as the time  shrinks to zero. Then, again informally, Z ∞ Z ∞ S(x − y, )f(y) dy → δ(x − y)f(y) dy = f(x) −∞ −∞ as  → 0.

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