Chapter 7 Infinite Series

7.6 Expand a function into the sine series and cosine series 3 The Fourier series of a function of period 2l 1 The Fourier series of odd function and even function 2 Expand a function into the sine or cosine series

1 The Fourier series of odd function and even function Theorem 1 The Fourier coefficients for an even or odd function have special forms

Proof odd function even function

Similarly, we can prove (2) Example 1 Solution

picture of sum function

Periodic Extensions 2 Expand a function into the sine or cosine series Odd Extensions

Even Extensions

Example 2 Solution (1) Find sine series

(2) Find cosine series

3 The Fourier series of a function of period 2l Theorem 2

Remark:

Example 3 Solution

Example 4 Solution

Example 5 Solution

or