Page 134 - Circuit Analysis II with MATLAB Applications

Chapter 4 The Laplace Transformation

*
The integral of (4.42) is an improper integral but converges (approaches a limit) for all n ! . 0
We will now derive the basic properties of the gamma function, and its relation to the well known
factorial function
–
n! = nn 1 n2 ˜ – ˜ 321

˜
˜

The integral of (4.42) can be evaluated by performing integration by parts. Thus, in (4.39) we let
u = e x – and dv = x n – 1
Then,
n
x
du = e – x – dx and v = -----
n
and (4.42) is written as

n – x f 1 f
n –
x
* n = x e + -- - ³ x e d x (4.43)
------------
n n
x = 0 0
With the condition that n ! 0 , the first term on the right side of (4.43) vanishes at the lower limit
x = 0 . It also vanishes at the upper limit as x o f . This can be proved with L’ Hôpital’s rule by dif-
ferentiating both numerator and denominator m times, where m t n . Then,

d m x n d m – 1 nx n – 1
n – x n m m – 1
x e x d x d x
lim ------------ = lim -------- = lim ------------------- = lim ------------------------------------ = }
x o f n x o f ne x x o f d m ne x x o f d m – 1 ne x
d x m d x m – 1
n – m
1
–
1 n –
n –
n –
m +

2 }

= lim nn – 1 n2 } – nm + 1 x lim -------------------------------------------------------------------- = 0
------------------------------------------------------------------------------------ =
–
x o f ne x x o f x mn x
e
Therefore, (4.43) reduces to
1 f n – x
* n = -- - ³ x e d x
n 0
and with (4.42), we have
* Improper integrals are two types and these are:
b
a. ³ fx x where the limits of integration a or b or both are infinite
d
a
b
b. ³ fx x where f(x) becomes infinite at a value x between the lower and upper limits of integration inclusive.
d
a
4-16 Circuit Analysis II with MATLAB Applications
Orchard Publications