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

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

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

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

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 -

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 -

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.

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

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

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

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

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

Now change the limits of integration.

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

This is an odd function. Therefore,

Use integration by parts.

when n is even.

Therefore, the Fourier series is

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

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

Denoting , and

The Fourier series for can be expressed as:

The coefficients can be evaluated in the following manner.

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

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

Thank You