1.1 Basic Concepts and Formulae
3
a n =
1
L
L
−L
f (x) cos(nπ x/L) dx
(1.5)
b n =
1
L
L
−L
f (x) sin(nπ x/L) dx
(1.6)
Complex form of Fourier series
Assuming that the Series (1.1) converges at f (x),
f (x) =
∞
n=−∞
C n e
inπ x/L
(1.7)
with
C n =
1
L
C+2L
C
f (x)e
−iπnx/L dx =
⎧
⎪ ⎨
⎪ ⎩
1
2
(a n − ib n ) n > 0
1
2
(a −n + ib −n ) n < 0
1
2
a o
n = 0
(1.8)
Fourier transforms
The Fourier transform of f (x) is defined as
( f (x)) = F(α) =
∞
−∞
f (x)e
iαx dx
(1.9)
and the inverse Fourier transform of F(α) is
−1 ( f (α)) = F(x) =
1
2π
∞
−∞
F(α)e
i∝x dα
(1.10)
f (x) and F(α) are known as Fourier Transform pairs. Some selected pairs are given
in Table 1.1.
Table 1.1
f (x)
F(α)
f (x)
F(α)
1
x 2 + a 2
π e
−aα
a
e
−ax
a
α 2 + a 2
x
x 2 + a 2
−
πiα
a
e
−aα
e
−ax 2
1
2
π
a
e
−α 2 /4a
1
x
π
2
xe
−ax 2
√ π
4a 3/2 αe
−α 2 /4a
Gamma and beta functions
The gamma function Γ(n) is defined by