Mathematical Methods for Physicists: A concise introduction - Site Map

LAPLACE TRANSFORMS OF ELEMENTARY FUNCTIONS
Solution:
We have to check whether or not
lim
x!1 ebx x 3 x 3
ˆ lim
x!1 e bx
exists. Now if b > 0, then L'Hospital's rule gives
lim
x!1 ebx x 3 ˆ lim
x!1
x 3
Therefore x 3 is of exponential order as x !1.
3x 2 6x 6
ˆ lim ˆ lim
bx
e x!1 bx
be x!1 b 2 ˆ lim
bx
e x!1 b 3 ˆ 0:
bx
e
Laplace transforms of some elementary functions
Using the de®nition (9.1) we now obtain the transforms of polynomials, exponential
and trigonometric functions.
(1) f …x† ˆ1 for x > 0.
By de®nition, we have
L‰1Š ˆ
Z 1
(2) f …x† ˆx n , where n is a positive integer.
By de®nition, we have
0
e px dx ˆ 1
p ; p > 0:
Using integration by parts:
Z
L‰x n Šˆ
Z 1
0
Z
uv 0 dx ˆ uv
e px x n dx:
vu 0 dx
with
u ˆ x n ; dv ˆ v 0 dx ˆ e px dx ˆ…1=p†d…e px †; v ˆ…1=p†e px ;
we obtain
Z 1
0
e px x n dx ˆ xn e px
p
1
0
‡ n p
Z 1
0
e px x n1 dx:
For p > 0 and n > 0, the ®rst term on the right hand side of the above equation is
zero, and so we have
Z 1
0
e px x n dx ˆ n
p
375
Z 1
0
e px x n1 dx