B - The delta function

Summary

As we have seen in Chapter 6 the δ function (sometimes called the Dirac delta function) is a useful mathematical tool. In this Appendix we derive formulae for the representation of the delta functions employed in Chapter 6.

The δ function is defined by its properties:

where f(x) and its derivative are continuous, single-valued functions and the integral is over any range containing x₀. The result, ∫ δ(x – x₀) dx = 1, follows from (B.2) for f(x) = 1. Another property, δ(x – x₀) → ∞ as x → x₀, is implied by (B.1) and (B.2). Clearly if (B.1) holds, the δ function must be arbitrarily large at x₀ if (B.2) is valid.

There are numerous analytical representations of the delta function. We shall use a frequently employed representation wherein δ(x – x₀) is the limit of a particular function:

where “ℑM” indicates the imaginary part of the quantity and λ is a small positive number. In using this representation there is an implied order of doing things. The limiting process λ → 0 (λ > 0) is to be performed last. This means one must calculate the imaginary part first, then take the limit as λ → 0. This limiting process is often indicated by using the symbol 0⁺ as we did in Chapter 6.

The imaginary part of (B.3) is

It is easy to show that the delta function defined by (B.4) satisfies the equations (B.1) and (B.2).