Mathematical Methods for Physicists: A concise introduction - Site Map

THE LAPLACE TRANSFORMATION
or
L‰x n Šˆn
p L‰xn1 Š
from which we may obtain for n > 1
L‰x n1 Šˆn 1 L‰x n2 Š:
p
Iteration of this process yields
n 1†…n 2†2 1
L‰x Šˆn…n
p n L‰x 0 Š:
By (1) above we have
L‰x 0 ŠˆL‰1Š ˆ1=p:
Hence we ®nally have
L‰x n Šˆ n! ; p > 0:
pn‡1 (3) f …x† ˆe ax , where a is a real constant.
Z 1
L‰e ax Šˆ e px e ax dx ˆ 1
0
p a ;
where p > a for convegence. (For details, see Example 9.1.)
(4) f …x† ˆsin ax, where a is a real constant.
L‰sin axŠ ˆ
Z 1
0
e px sin axdx:
Using Z Z
uv 0 dx ˆ uv
vu 0 dx with u ˆ e px ; dv ˆd…cos ax†=a;
and
Z
e mx sin nxdx ˆ emx …m sin nx n cos nx†
n 2 ‡ m 2
(you can obtain this simply by using integration by parts twice) we obtain
L‰sin axŠ ˆ
Z 1
0
e px sin axdx ˆ
epx …p sin ax a cos ax
p 2 ‡ a 2
Since p is positive, e px ! 0asx !1, but sin ax and cos ax are bounded as
x !1, so we obtain
1…0 a†
L‰sin axŠ ˆ0
p 2 ‡ a 2 ˆ a
p 2 ‡ a 2 ; p > 0:
1
0
:
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