MATH528

MATH528 Lesson16: Heaviside step function, Dirac delta function

Definition. The Heaviside (step) function $u (t - a)$ is defined as

$u (t - a) = {\begin{cases} 0 & if t < a \\ 1 & if t > a \end{cases} . . (a ⩾ 0), ℒ [u (t - a)] = \frac{e^{- a s}}{s}$

Definition. The impulse of force $f (t)$ over time interval $[a, a + k]$ is the integral

$I = \int_{a}^{a + k} f (t) d t .$

Consider $k$ small such that $f (t)$ is approximately constant and described by

f_{k} (t - a) = {\begin{cases} 1 / k & a ⩽ t ⩽ a + k \\ 0 & otherwise \end{cases} ., I_{k} = \int_{a}^{a + k} f_{k} (t) d t = 1 .

Note that $f_{k} (t - a) = \frac{1}{k} [u (t - a) - u (t - a - k)]$ such that

F_{k} (s) = ℒ (f_{k}) (s) = \frac{1}{k} [\frac{e^{- a s}}{s} - \frac{e^{- (a + k) s}}{s}], {lim}_{k \to 0} F_{k} (s) = e^{- a s} = ℒ [δ (t - a)]

Definition. The (generalized) function ${lim}_{k \to 0} f_{k} (t) = δ (t - a) = ℒ^{- 1} [e^{- a s}]$ is the Dirac delta function.

Properties:

\begin{array}{ll} \int_{0}^{\infty} δ (t - a) d t = 1 & \int_{0}^{\infty} g (t) δ (t - a) d t = g (a) \end{array}