12.1 The Dirichlet conditions:

1. Chapter 12 Fourier series • Advantages: • describes functions that are not everywhere continuous and/or differentiable. • represent the response of a system to a period input and depend on the frequency of the input • using in string vibration, light scattering, input signal transmission in electronic circuit 12.1 The Dirichlet conditions: • The function must be periodic. • It must be single-valued and continuous, except possibly at a finite number of finite discontinuities. • It must have only a finite number of maxima and minima within one period. • The integral over one period of a function must converge.

2. Chapter 12 Fourier series • all functions may be written as the sum of an odd and an even part chosen as the sum of a cosine series chosen as the sum of a sine series • orthogonal properties: the length of a period is L:

3. Chapter 12 Fourier series • Fourier series expansion of the function f(x) is

4. Chapter 12 Fourier series Ex: Express the square-wave function as a Fourier series • f(t) is an odd function, so only the sine term survives

5. Chapter 12 Fourier series 12.4 Discontinuous functions (1) At a point of finite discontinuity, , the Fourier series converges to (2) At a discontinuity, the Fourier series representation of the function will overshoot its value. It never disappears even in the limit of an infinite number of terms. This behavior is known as Gibb’s phenomenon. 2 terms 1 term overshooting 3 terms 20 terms

6. Chapter 12 Fourier series 12.5 Non-periodic functions: period=L, no particular symmetry period=2L, antisymmetry; odd fun period=2L, symmetry; even fun

7. Chapter 12 Fourier series Ex. : Find the Fourier series of (1) make the function periodic and symmetric

8. Chapter 12 Fourier series (2) make the function periodic and antisymmetric

9. Chapter 12 Fourier series • Integration and differentiation (1) The Fourier series of f(x) is integrated term by term then the resulting Fourier series converges to the integral of f(x). (2) f(x) is a continuous function of x and is periodic then the Fourier series that results from differentiating term by term converges to f(x). Ex: Find the Fourier series of Sol: from the previous example integrate (1) term by term integrate (1) term by term put (3) into (2)

10. Chapter 12 Fourier series 12.7 Complex Fourier series Complex Fourier series expansion is:

11. Chapter 12 Fourier series Ex: Find a complex Fourier series for in the range

12. Chapter 12 Fourier series 12.8 Parseval’s theorem: • general proof:

13. Chapter 12 Fourier series Ex: Using Parseval’s theorem and the Fourier series for evaluate the sum