Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

313

Properties of even/odd functions

a) If a function is even, then

ℓ Z

ℓ Z 0

f ( x ) d x = 2

f ( x ) d x .

− ℓ

b) If a function is odd, then

ℓ Z

f ( x ) d x = 0 .

− ℓ

c) A product of even functions p ( x )= f ( x ) · g ( x ) is an even function p ( − x )= f ( − x ) · g ( − x )= f ( x ) · g ( x )= p ( x ) . d) A product of odd functions is an even function p ( − x )= f ( − x ) · g ( − x )=( − 1 ) f ( x ) · ( − 1 ) g ( x )= f ( x ) · g ( x )= p ( x ) . e) A product of an even function f ( x ) and an odd function g ( x ) is an odd function p ( − x )= f ( − x ) · g ( − x )= f ( x ) · ( − 1 ) g ( x )= − f ( x ) · g ( x )= − p ( x ) . These properties are used in calculating the Fourier series coefficients a k and b k , when f ( x ) is an even or odd function. Namely, as the product of f ( x ) and sin x (sin x is an odd function), or cos x (cos x is an even function) appear under integral (6.12), then, depending on f ( x ) and the given the properties a)-d), some coefficients can be equal to zero. Fourier cosine series and Fourier sine series Let f ( x ) be an even function. Then p ( x ) = f ( x ) cos x is an even function, and q ( x ) = f ( x ) sin x is an odd function. Using the properties of these function, we obtain the follow ing Fourier coefficients

1 ℓ Z 1 ℓ Z 1 ℓ Z

2 ℓ Z

ℓ

ℓ

a 0 =

f ( x ) d x =

f ( x ) d x ,

0

− ℓ

2 ℓ Z

k π x ℓ

k π x ℓ

ℓ

ℓ

a k =

f ( x ) cos

d x =

f ( x ) cos

d x ,

(6.16)

0

− ℓ

k π x ℓ

ℓ

b k =

f ( x ) sin

d x = 0 .

− ℓ

In this special case, the Fourier series takes the form

∞ ∑ k = 1

k π ℓ

a 0 2

f ( x )=

a k cos

x .

(6.17)

+

In this case it is said that the function is expanded into a Fourier cosine series , where coefficients a k are determined by relations (6.16). Let f ( x ) be an odd function. Then p ( x )= f ( x ) cos x is an odd function, and q ( x )= f ( x ) sin x is an even function. Using the properties of these function, we obtain the following Fourier