A function, f(x), is an even function if f(x) = f(−x) and is an odd
function if f(x) = −f(−x). When a function, f(x), is neither even
nor odd, it can be decomposed into the even part and the odd part as
f(x) = G(x) + H(x),
(1)
where
G(x)
=
f(x) + f(−x)
2
,
(2)
H(x)
=
f(x) − f(−x)
2
.
(3)
The function, G(x), is the even part of f(x) and the function,
H(x), is the odd part of f(x). Example 1: f(x) = ex: G(x)=(ex + e−x)/2 = cosh x and H(x) = (ex − e−x)/2 = sinh x. The even part of ex is the hyperbolic cosine function and
the odd part of ex is the hyperbolic sine function. Example 2: f(x) = eix: G(x)=(eix + e−ix)/2 = cos x and H(x) = (eix − e−ix)/2 = sin x.
The even part of eix is the cosine function and the odd part
of eix is the sine function.
Similarities:
cos2x + sin2x = 1, cosh2x − sinh2x = 1,
(cos x)′ = − sin x, (cosh x)′ = sinh x,
(sin x)′ = cos x, (sinh x)′ = cosh x.
Fourier Series
The function, eimx, is called the fundamental periodic function
as it is a combination of sin mx and cos mx. A periodic
function, f(x), over [−π, π] can be expanded by a linear
combination of the fundamental periodic functions as
f(x) =
∞ ∑ m=−∞
cmeimx.
(4)
Multiplying e−inx on the both sides of Eq.(4) and
integrating the result from −π to π yields
⌠ ⌡
π
−π
f(x) e−inxdx
=
∞ ∑ m=−∞
cm
⌠ ⌡
π
−π
ei(m−n)xdx
=
2πcn,
(5)
thus
cn =
1
2π
⌠ ⌡
π
−π
f(x) e−inxdx,
(6)
where
⌠ ⌡
π
−π
eimxe−inxdx =
⎧ ⎪ ⎨
⎪ ⎩
0
m ≠ n
2π
m=n
was used.
Equations (4, 6) can be converted into real format
as