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):
simplifyDirac(Dirac(x,1)*x) |
(x−2)3δ(3)(1−x/2) simplifies to:
simplifyDirac((x-2)^3*Dirac(1-x/2,3)) |
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:
simplifyDirac(sin(1-cos(2x))*Dirac(x,2)) |
simplifyDirac(tan(x-pi/2)*Dirac(x,2)) |
If f(x)=p(x)/q(x) is rational with p(x)=(x−x1)(x−x2)⋯(x−xn), xi≠ xj
for i≠ j, and q(x)≠ 0 ∀ x∈ℝ, then δ(f(x))
expands to ∑k=1nδ(x−xk)/f′(xk). For example:
simplifyDirac(Dirac(x^2-x-6)) |