Show that the family of functions (x) with: (x) = 1 2 e |x|/ tend to (x)

Show that the family of functions φ(x) with:
φ(x) = 1
2
e
−|x|/
tend to δ(x) in the limit  → 0, by explicitly calculating the integral of φ(x) over a smooth test function
f(x) and then taking the limit  → 0.
Hint(s): You may expand the function f(x) as a Taylor series around x = 0 and will require to evaluate
the integral R ∞
−∞ dx 1
2
e
−|x|/x
n
for even integer values n.

Answer rating: 100% (QA)

As 0 we have x 1 2 e x 1 2 e x The integra... View the full answer