Page 133 - Circuit Analysis II with MATLAB Applications

The Laplace Transform of Common Functions of Time

t – e – st f f e – – st t – e – st e – st f
L t^` = ------------- – ³ -----------dt = ------------- – -------- (4.40)
s s s 2
0 0 s
0
Since the upper limit of integration in (4.40) produces an indeterminate form, we apply L’ Hôpital’s
*
rule , that is,
d t
1
t
t d
lim te – st = lim ------ = lim ---------------- = lim -------- = 0
t o f t o f e st t o f d e st t o f se st
t d
1
Evaluating the second term of (4.40), we get L t^` = ----
s 2
Thus, we have obtained the transform pair

1
t œ ---- (4.41)
s 2

for V ! . 0

Example 4.3
n
Find L t u t ^ 0 `

Solution:
To find the Laplace transform of this function, we must first review the gamma or generalized facto-

rial function * n defined as

f
e d
* n = ³ x n – 1 – x x (4.42)
0

fx
* Often, the ratio of two functions, such as ---------- , for some value of x, say a, results in an indeterminate form. To
gx
fx
work around this problem, we consider the limit lim ---------- , and we wish to find this limit, if it exists. To find this
x o a gx
d d
limit, we use L’Hôpital’s rule which states that if f a = ga = 0 , and if the limit ------fx ------gx e as x
dx dx
d
fx
approaches a exists, then, lim ---------- = lim § ------fx ------gx e d ·
x o a gx x o a © dx dx ¹

Circuit Analysis II with MATLAB Applications 4-15
Orchard Publications