1. Fourier Series

2. Fourier Analysis Discrete Continuous Fourier Series Fourier Integral
Fast
Fourier
Transform
Discrete
Fourier
Transform
Fourier
Transform

3. Outline Periodic Function Fourier Cosine and Sine Series
Periodic Function with Period 2L
Odd and Even Functions
Half Range Fourier Cosine and Sine Series
Complex Notation for Fourier Series

4. Fourier, Joseph
Fourier, Joseph

5. Fourier, Joseph
In 1807, Fourier submitted a paper to the Academy of Sciences of Paris. In it he derived the heat equation and proposed his separation of variables method of solution. The paper, evaluated by Laplace, Lagrange, and Lagendre, was rejected for lack rigor. However, the results were promising enough for the academy to include the problem of describing heat conduction in a prize competition in Fourier’s 1811 revision of his earlier paper won the prize, but suffered the same criticism as before. In 1822, Fourier finally published his classic Theorie analytique de la chaleur, laying the fundations not only for the separation of variables method and Fourier series, but for the Fourier integral and transform as well.

6. Periodic Function Definition: Periodic Function
A function f(x) is said to be periodic with period T if for all x
f(x) x T

7. Periodic Function f(x+p)=f(x), f(x+np)=f(x)
If f(x) and g(x) have period p, the the function H(x)=af(x)+bg(x) , also has the period p
If a period function of f(x) has a smallest period p (p >0), this is often called the fundamental period of f(x)

8. Periodic Function Example Cosine Functions: cosx, cos2x, cos3x, …
Sine Functions: sinx, sin2x, sin3x, …
eix, ei2x, ei3x, …
e-ix, e-i2x, e-i3x, …

9. Fourier Cosine and Sine Series
Lemma: Trigonometric System is Orthogonal

10. Fourier Cosine and Sine Series
A function f(x) is periodic with period 2 and

11. Fourier Cosine and Sine Series (Euler formulas)
Then

12. Fourier Cosine and Sine Series
Proof:

13. Fourier Cosine and Sine Series
Example 5-1: Find the Fourier coefficients corresponding to the function
Sol:

14. Fourier Cosine and Sine Series
Sol:

15. Fourier Cosine and Sine Series
Sol:

16. Periodic Function with Period 2L
A periodic function f(x) with period 2L
f(x) x 2L

17. Periodic Function with Period 2L
Then

18. Periodic Square Wave

19. Odd and Even Functions A function f(x) is said to be even if
A function f(x) is said to be odd if

20. Odd and Even Functions
Even Function
Odd Function
f(x)
f(x) x x

21. Odd and Even Functions Property
The product of an even and an odd function is odd.

22. Odd and Even Functions
Fourier Cosine Series
Fourier Sine Series

23. Sun of Functions
The Fourier coefficients of a sum f1+f2 are the sum of the corresponding Fourier coefficients of f1 and f2.
The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.

24. Examples Rectangular Pulse Sawtooth wave
The function f*(x) is the sum in Example 1 of Sec.10.2 and the constant k.
Sawtooth wave
Find the Fourier series of the function

25. Half-Range Expansions
A function f is given only on half the range, half the interval of periodicity of length 2L.
even periodic extention f1 of f
odd period extention f2 of f

26. Complex Notation for Fourier Series

27. Complex Notation for Fourier Series
A periodic function f(x) with period 2L

28. Complex Fourier Series
Find complex Fourier series

29. Exercise Section 10-4 Section 10-2 Section 10-3 #1, #5, #11 #5, #9
#1, #15

30. Fourier Cosine and Sine Integrals
Example 1

31. Fourier Cosine and Sine Integrals
sin(x) x

32. Fourier Cosine and Sine Integralsx

33. Fourier Cosine and Sine Integrals

34. Fourier Cosine and Sine Integrals

35. Fourier Cosine and Sine Integrals

36. Gibb’s Phenomenon
Sine Integral

37. Gibb’s Phenomenon
Gibb’s Phenomenon

38. Gibb’s Phenomenon

39. Fourier Cosine and Sine Integrals
Fourier Cosine Integral of f(t)

40. Fourier Cosine and Sine Integrals
Fourier Sine Integral of f(t)

41. Fourier Integrals
Fourier Integral of f(t)