Homework Set 8 Math/ECE 430 due Friday, April 5, 2013

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```Homework Set 8
Math/ECE 430
due Friday, April 5, 2013
The page numbers below refer to the book by Gasquet and Witomski.
R∞
1. Show that hδ, φi = φ(0) implies that −∞ δ(x − y)φ(x)dx = φ(y)
2. Show that
Rb
a
δ(x − y)dx = 1 provided a &lt; y &lt; b ; otherwise
3. Show that δ(ax) =
Rb
a
δ(x − y)dx = 0.
1
δ(x). In particular, δ(−x) = δ(x).
|a|
4. Show that δ(g(x)) =
X δ(x − xj )
j
|g 0 (xj )|
, where xj are the roots of g(x). Here assume that
g 0 is nonzero.
5. Show that H 0 = δ, where the prime denotes the weak or distributional derivative and
H denotes the Heaviside function.
6. hδ 0 , φi =?
7. # 28.11 (solution of xT = 0 in D0 )
(a) Show that a function χ ∈ D satisfies χ(0) = 0 if and only if χ = xψ with ψ ∈ D.
(Hint: think about Taylor expansions with remainder).
(b) Let ζ ∈ D satisfy ζ(0) = 1. Show that for any φ ∈ D, there is some ψ ∈ D0 so
that φ can be written as φ = φ(0)ζ + xψ. (Hint: You just need to show that
χ = φ − φ(0)ζ is in D and is zero at the origin; then apply the above result. )
(c) Deduce from this that T ∈ D0 solves xT = 0 if and only if T = Kδ for some
constant K.
8. # 28.12 (solution of xT = 1 in D0 .) Deduce from the above exercise that T solves
xT = 1 if and only if T is of the form P.V.(1/x) + Kδ for some constant K. (Hint:
We know that P.V.(1/x) is a solution; subtract it off to obtain the type of problem
treated in problem 7.)
1
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