Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

321

We have thus shown how coefficients a k are computed for an arbitrary period ℓ . In a similar way, the expression for b k can be obtained. It can be easily proved that the relation (6.32) also stands for an arbitrary period 2 ℓ , i.e.

ℓ Z

∞ ∑ k = 1

a 2 0 2

1 ℓ

2 k .

f 2 ( x ) d x =

a 2

k + b

(6.33)

+

− ℓ

This relation (6.33) is known as Parseval’s 6 identity for the Fourier series.

R Note, that if the function is periodic in interval ( a , b ) , then, as in the previous case, we obtain (for period b − a ) f ( x )= a 0 2 + ∞ ∑ k = 1 a k cos 2 π k b − a ( x − a )+ b k sin 2 π k b − a ( x − a ) ,

b Z a b Z a

2 π k b − a 2 π k b − a

2 b − a 2 b − a

a k =

f ( x ) cos

( x − a ) d x ,

b k =

f ( x ) sin

( x − a ) d x .

6.2.5 Complex form of Fourier series

Using Euler formulas for complex numbers

e k π x

e k π x

k π x

k π x

ℓ i + e −

ℓ i

ℓ i − e −

ℓ i

k π x ℓ

k π x ℓ

cos

sin

=

,

=

,

2

2 i

we can represent the Fourier series of the function f ( x ) in the form

∞ ∑ k = 1 − 1 ∑ − ∞

∞ ∑ k = 1

a 0 2 a 0 2

a k − b k i 2 a k + b k i 2

a k + b k i 2

k π x

k π x

ℓ i +

ℓ i =

e −

f ( x )=

e

+

∞ ∑ 1

a k − b k i 2

k π x

k π x

ℓ i +

ℓ i =

e

e

+

=

∞ ∑ − ∞

k π x

ℓ i .

c k e

=

Here, we have

    

a k − b k i 2

k > 0 ,

a 0 2

k = 0 ,

c k =

a k + b k i 2

k < 0 ,

where c k can be determined by the relation

ℓ Z

1 2 ℓ

k π x ℓ i d x .

f ( x ) e −

c k =

− ℓ

6 Marie Antoine Parseval (1755-1836), famous French mathematician.