10.1.6  Simplification of expressions involving Dirac delta distribution

The simplifyDirac command simplifies expressions involving Dirac delta distribution (see Section 7.3.10).

• simplifyDirac takes one mandatory argument and a sequence of optional arguments:
  • expr, an expression.
  • Optionally, x, a variable with respect to which the simplification is done (by default, x=x).
• simplifyDirac(expr ⟨,x ⟩) returns a simplified form of expr as a linear combination of Dirac delta and its derivatives, with coefficients being simplified by using simplify.

Examples

Since δ(x)=−x δ′(x):

(x−2)³δ⁽³⁾(1−x/2) simplifies to:

By using the identity xⁿδ⁽ⁿ⁾(x)=(−1)ⁿn! δ(x), x δ(x)=0 and Laurent series expansions, expressions of the form f(x) δ⁽ⁿ⁾(x−a) can be written as linear combination of delta function and its derivatives with constant coefficients. For example, with f(x)=tan(x−π/2), a=0 and n=2:

If f(x)=p(x)/q(x) is rational with p(x)=(x−x₁)(x−x₂)⋯(x−x_n), x_i≠ x_j for i≠ j, and q(x)≠ 0 ∀ x∈ℝ, then δ(f(x)) expands to ∑_k=1ⁿδ(x−x_k)/f′(x_k). For example: