1000 Solved Problems in Modern Physics

8 1 Mathematical Physics

Laplace transforms:

Definition:
A Laplace transform of the functionF(t)is

∫∞

0

F(t)e−stdt=f(s) (1.55)

The functionf(s) is the Laplace transform ofF(t). Symbolically,L{F(t)}=
f(s) andF(t)=L−^1 {f(s)}is the inverse Laplace transform off(s).L−^1 is called
the inverse Laplace operator.

Table of Laplace transforms:

F(t) f(s)

aF 1 (t)+bF 2 (t) af 1 (s)+bf 2 (s)

aF(at) f(s/a)

eatF(t) f(s−a)

F(t−a)t>a

0 t<a e

−asf(s)

1

1

s

t

1

s^2

tn−^1

(n−1)!

1

sn

n= 1 , 2 , 3 ,...

eat

1

s−a

sinat

a

1

s^2 +a^2

cosat

s

s^2 +a^2

sinhat

a

1

s^2 −a^2

coshat

s

s^2 −a^2

Calculus of variation

The calculus of variation is concerned with the problem of finding a functiony(x)
such that a definite integral, taken over a function shall be a maximum or minimum.
Let it be desired to find that functiony(x) which will cause the integral

I=

∫x 2

x 1

F(x,y,y′)dx (1.56)