SET II : The Dirac Delta Function

1.

  Consider the function f(x),

 

$$\left\{\begin{array}{cc} 0 & \mbox{\hspace{1cm} $\mid x \mid > a/2$ } \\ 1/a & \mbox{\hspace{1cm} $\mid x \mid \leq a/2$ } \end{array} \right.$$ f(x) = (1)

(a)

Plot f(x) for a = 1 and a = 1/2.

(b)

Evaluate $\int_{- \infty}^{\infty} f(x) \, dx$.

2.

The Dirac delta-function $\delta (x)$ is defined as

 

$$\delta (x) \lim_{a \rightarrow 0} f (x)$$ = (2)

where f(x) is defined in problem  above. Obviously $\delta (x = 0)$ is undefined. Evaluate

(a)

$\delta (x \neq 0)$.

(b)

$\int_{- b}^{b} \delta (x) \,dx$, where b is a real positive constant.

(c)

$\int_{- b}^{b} \delta (- x) \, dx$, where b is a real positive constant.

(d)

$\int_{c - b}^{c + b} \delta (x - c) \, dx$, where b and c are real positive constants.

3.

Let g (x) be a smooth function, and `b' be a real constant. Show that

(a)

$$\int_{- \infty}^{\infty} \delta (x - b) g(x) \,dx$$ = g(b) (3)

(b)

$$\int_{- \infty}^{\infty} \delta (bx) g(x) \,dx \frac{1}{\mid b \mid} g(0)$$ = (4)

4.

OPTIONAL : If g(x) is a smooth function with the zero given by g(x₀) = 0, show that

 

$$\delta (g(x)) \frac{\delta (x - x_0)}{\left\vert \frac{d}{dx} g(x_0) \right\vert }$$ = (5)

Using Eqn () show that

$$\delta (x^2 - a^2) \frac{1}{2 \vert a\vert} [\delta (x - a) + \delta (x - a)]$$ = (6)

©Vijay A. Singh