Mathematical Methods for Physicists: A concise introduction - Site Map

LAPLACE TRANSFORMS OF ELEMENTARY FUNCTIONS
(5) f …x† ˆcos ax, where a is a real constant.
Using the result
Z
e mx cos nxdx ˆ emx …m cos nx ‡ n sin mx†
n 2 ‡ m 2 ;
we obtain
L‰cos axŠ ˆ
Z 1
0
e px cos axdx ˆ
(6) f …x† ˆsinh ax, where a is a real constant.
p
p 2 ‡ a 2 ; p > 0:
Using the linearity property of the Laplace transform operator L, we obtain
L‰cosh axŠ ˆL eax ‡ e ax
ˆ 1
2 2 L‰eax Š‡ 1 2 L‰eax Š
ˆ 1
1
2 p a ‡ 1
p
ˆ
p ‡ a p 2 a 2 :
(7) f …x† ˆx k , where k > 1.
By de®nition we have
L‰x k Šˆ
Z 1
0
e px x k dx:
Let px ˆ u, then dx ˆ p 1 du; x k ˆ u k =p k , and so
Z 1
L‰x k Šˆ e px x k dx ˆ 1 Z 1
p k‡1 u k e u du ˆ
0
0
…k ‡ 1†
p k‡1 :
Note that the integral de®ning the gamma function converges if and only if
k > 1.
The following example illustrates the calculation of inverse Laplace transforms
which is equally important in solving di€erential equations.
Example 9.3
Find
…a† L 1 5
; …b† L 1 1
p ‡ 2
p s ; s > 0:
Solution:
…a† L 1 5
p ‡ 2
ˆ 5L 1 1
:
p ‡ 2
377