Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

330

Problem 230 Expand the following function into a Fourier series f ( x )= A 0 < x < l , 0 l < x < 2 l ,

in the interval ( 0 , 2 l ) , where A is a constant.

Result

∞ ∑ k = 0

( 2 k + 1 ) π x l .

A 2

2 A π

1 2 k + 1

f ( x )=

sin

+

Problem 231 Expand the function f ( x )= π 2 − x 2 into a Fourier series in the interval ( − π , π ) .

Result

∞ ∑ n = 1

( − 1 ) n + 1 n 2

2 3

π 2 + 4

cos nx .

f ( x )=