8
1 Mathematical Physics
Laplace transforms:
Definition:
A Laplace transform of the function F(t) is
∞
0
F(t)e
−st dt = f (s)
(1.55)
The function f (s) is the Laplace transform of F(t). Symbolically, L{F(t)} =
f (s) and F(t) = L
−1
{ f (s)} is the inverse Laplace transform of f (s). L
−1 is called
the inverse Laplace operator.
Table of Laplace transforms:
F(t)
f (s)
a F 1 (t) + bF 2 (t)
a f 1 (s) + b f 2 (s)
a F(at)
f (s/a)
e
at F(t)
f (s − a)
F(t − a) t > a
0
t < a
e
−as f (s)
1
1
s
t
1
s 2
t
n−1
(n − 1)!
1
s n n = 1, 2, 3, . . .
e
at
1
s − a
sin at
a
1
s 2 + a 2
cos at
s
s 2 + a 2
sinh at
a
1
s 2 − a 2
cosh at
s
s 2 − a 2
Calculus of variation
The calculus of variation is concerned with the problem of finding a function y(x)
such that a definite integral, taken over a function shall be a maximum or minimum.
Let it be desired to find that function y(x) which will cause the integral
I =
x 2
x 1
F(x, y, y
) dx
(1.56)