Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

316

Example 216 Expand the following function into a Fourier series f ( x )=    2 k ℓ x , 2 k ℓ ( ℓ − x ) , ℓ 2

ℓ 2

0 < x <

,

< x <ℓ.

Solution Let us first sketch the given function.

Figure 6.4

According to the conditions of the Fourier series convergence theorem, the function needs to be periodic. Therefore, we will extend the initial function to: a) even and b) odd periodic function. a) Let us now sketch the extended function, according to the instructions given earlier.

Figure 6.5

In this case (even function), according to (6.16), we obtain

1 ℓ   2 ℓ  

( ℓ − x ) d x   ,

ℓ/ 2 Z 0 ℓ/ 2 Z 0

ℓ Z ℓ/ 2

a 0 2

2 k ℓ

2 k ℓ

x d x +

=

x d x 

ℓ Z ℓ/ 2

n π ℓ

n π ℓ

2 k ℓ

2 k ℓ

 ,

a n =

x cos

x d x +

( ℓ − x ) cos

b n = 0 .