Integral Transformations - Heat Conduction: Mathematical Models and Analytical Solutions - page 447

Thus

2 2

a 0
2 =

1
T

1
T

e − i n ω t d t c 0 = g ( t ) d t , c n = g ( t ) ,

2 2

− −

T
2

1
T

e i n ω t d t

c − n = g ( t ) ,

2

−

or

2

1
T

e − i n ω t d t c n = g ( t ) , n = 0 ,± 1 ,± 2 , ···.

T
2

−

Finally we obtain the Fourier series of exponential form

1
∑
n = − ∞

− + ∞

n = 1 c − n e i n ω t

c n e i n ω t

g ( t )= c 0 + +

e − iω n t d t e iω n t

T
2
−

+ ∞
∑

+ ∞
∑

1
T

c n e i n ω t = = g ( t ) , (B.1)

2

n = − ∞ n = − ∞

where ω n = n ω .

Fourier Integral

Consider a non-periodic function f ( t ) . Introduce a periodic function f T ( t ) of period

T such that

T
2 ,

T
2

f T ( t )= f ( t ) , t ∈ − .

( )= ( ) .

Therefore f t lim f T t

T → + ∞

Let Δω = ω − ω

1 . Thus

n n −

2 T 1

T =

1
2

Δω = n ω − ( n − 1 ) ω = or

π Δω .

Since Δω → 0as T → + ∞ , we have, by Eq. (B.1) and when f ( t ) satisfies some

conditions,

e iω n t

T
2

+ ∞
∑

1
2

e − iω n τ d f ( t )= lim f T ( t )= lim

Δω →

f T ( τ ) τ Δω

π

T → + ∞

2

−

n = − ∞

0

( T → + ∞ )

+ ∞ e iω t d

+ ∞

1
2

e − iωτ d

= f ( τ ) τ ω . (B.2)

π − ∞ − ∞