Delta functions live under integral signs, and two expressions (D1(x) and D2(x)) involving delta functions are said to be equal if ∫-∞^+∞ f(x) D1(x) dx = ∫-∞^+∞ f(x) D2(x) dx, for every (ordinary) function f(x) . (a) Show that δ(cx) = 1/c δ(x),where c is a real constant. (Be sure to check the case

Question

Delta functions live under integral signs, and two expressions [latex] (D_1(x)[/latex] and [latex] D_2(x)[/latex]) involving delta functions are said to be equal if

[latex] \int_{-\infty}^{+\infty} f(x) D_{1}(x) d x=\int_{-\infty}^{+\infty} f(x) D_{2}(x) d x [/latex],

for every (ordinary) function f(x) .

(a) Show that

[latex] \delta(c x)=\frac{1}{|c|} \delta(x) [/latex], (2.145)

where c is a real constant. (Be sure to check the case where c is negative.)

(b) Let θ(x) be the step function:

[latex] heta(x) \equiv \begin{cases}1, & x>0 \ 0, & x<0\end{cases} [/latex] (2.146).

(In the rare case where it actually matters, we define θ(0) to be 1/2.) Show that dθ/dx = δ(x).

Accepted Answer

(a) Let [latex] y \equiv c x [/latex] so [latex] d x=\frac{1}{c} d y .\left\{\begin{array}{l} ext { If } c>0, y:-\infty ightarrow \infty \ ext { If } c<0, y: \infty ightarrow-\infty \end{array} ight\} [/latex].

[latex] \int_{-\infty}^{\infty} f(x) \delta(c x) d x=\left\{\begin{array}{l} \frac{1}{c} \int_{-\infty}^{\infty} f(y / c) \delta(y) d y=\frac{1}{c} f(0) \quad(c>0) ; ext { or } \ \frac{1}{c} \int_{\infty}^{-\infty} f(y / c) \delta(y) d y=-\frac{1}{c} \int_{-\infty}^{\infty} f(y / c) \delta(y) d y=-\frac{1}{c} f(0)(c<0) \end{array} ight. [/latex].

In either case, [latex] \int_{-\infty}^{\infty} f(x) \delta(c x) d x=\frac{1}{|c|} f(0)=\int_{-\infty}^{\infty} f(x) \frac{1}{|c|} \delta(x) d x . ext { So } \delta(c x)=\frac{1}{|c|} \delta(x) [/latex].

(b)

[latex] \int_{-\infty}^{\infty} f(x) \frac{d heta}{d x} d x=\left.f heta ight|_{-\infty} ^{\infty}-\int_{-\infty}^{\infty} \frac{d f}{d x} heta d x [/latex], (integration by parts).

[latex] =f(\infty)-\int_{0}^{\infty} \frac{d f}{d x} d x=f(\infty)-f(\infty)+f(0)=f(0)=\int_{-\infty}^{\infty} f(x) \delta(x) d x [/latex].

So dθ/dx = δ(x). [Makes sense: The θ function is constant (so derivative is zero) except at x = 0, where the derivative is infinite.].

Question 2.23: Delta functions live under integral signs, and two expressio...

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