1. Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28
Department of Mathematics Power Point Presentation Topic –Fourier Series Prof. Darekar S.R

2. Fourier Series be a periodic function with period
The function can be represented by a trigonometric series as:

3. What kind of trigonometric (series) functions are we talking about?

4. Even and Odd Functions
(We are not talking about even or odd numbers.)

5. Even Functions
The value of the function would be the same when we walk equal distances along the X-axis in opposite directions.
Mathematically speaking -

6. Odd Functions
The value of the function would change its sign but with the same magnitude when we walk equal distances along the X-axis in opposite directions.
Mathematically speaking -

7. The Fourier series of an even function
is expressed in terms of a cosine series.
The Fourier series of an odd function
is expressed in terms of a sine series.

8. Example 2. Find the Fourier series of the following periodic function.

10. Use integration by parts. Details are shown in your class note.

12. This is an even function.
Therefore,
The corresponding Fourier series is

13. Functions Having Arbitrary Period
w is the angular velocity in radians per second.

14. f is the frequency of the periodic function,
where
Therefore,

15. Now change the limits of integration.

19. Example 4. Find the Fourier series of the following periodic function.

20. This is an odd function. Therefore,

21. Use integration by parts.

22. when n is even.

23. Therefore, the Fourier series is

24. The Complex Form of Fourier Series
Let us utilize the Euler formulae.

25. The
th harmonic component of (1) can be
expressed as:

26. Denoting ,
and

28. The Fourier series for
can be expressed as:

29. The coefficients can be evaluated in the following manner.

31. is the complex conjugate of
Note that
is the complex conjugate of .
Hence we may write that .

32. The complex form of the Fourier series of
with period
is:

33. Thank You