Mathematical Methods for Physicists: A concise introduction - Site Map

SHIFTING (OR TRANSLATION) THEOREMS
Example 9.4
Show that:
…a† Le ‰ ax x n n!
Š ˆ
…p ‡ a† n‡1 ; p > a;
…b† Le ax b
‰ sin bxŠ ˆ
…p ‡ a† 2 ‡ b ; p > a:
2
Solution:
(a) Recall
L‰x n Šˆn!=p n‡1 ; p > 0;
the shifting theorem then gives
L‰e ax x n n!
Šˆ
…p ‡ a† n‡1 ; p > a:
(b) Since
a
L‰sin axŠ ˆ
p 2 ‡ a 2 ;
it follows from the shifting theorem that
L‰e ax b
sin bxŠ ˆ
…p ‡ a† 2 ‡ b ; p > a:
2
Because of the relationship between Laplace transforms and inverse Laplace
transforms, any theorem involving Laplace transforms will have a corresponding
theorem involving inverse Lapace transforms. Thus
If L 1 ‰ F…p† Š ˆ f …x†; then L 1 ‰ F…p a† Š ˆ e ax f …x†:
The second shifting theorem
This second shifting theorem involves the shifting x variable and states that
Given L‰ f …x†Š ˆ F…p†, where f …x† ˆ0 for x < 0; and if g…x† ˆf …x a†,
then
L‰g…x†Š ˆ e ap L‰ f …x†Š:
To prove this theorem, let us start with
from which it follows that
F…p† ˆLf…x† ‰ Š ˆ
e ap F…p† ˆe ap Lf…x† ‰ Š ˆ
Z 1
0
Z 1
0
e px f …x†dx
e p…x‡a† f …x†dx:
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