In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this part we calculate the Fourier series for some given functions . We also define Fourier series for functions

**Examples of Fourier series**

* Example (sawtooth wave function)* Find the Fourier series of the function

Further we define for

The graph of the function can be found here. We calculate the Fourier coefficients for . Notice that is an odd function, that is, and is an even function. Hence is an odd function for all Therefore . (You can convince yourself of this fact by doing the integral.)

To calculate for , notice that is an odd function, hence is even and therefore

The Fourier series is therefore given by

An illustration how the partial sums of the Fourier series

approximate the function can be found here and here.

The oscillations of around any jump discontinuity of are known as the Gibbs phenomenon.

**Question** In the sawtooth example: to which values do the partial sums converge as ? (See the theorem on pointwise convergence from the previous post.)

* Exercise(rectangular wave function)* Find the Fourier series of the function

and for all .

**Question** Do we have

?

Provide an argument for your answer. (See the theorem on pointwise convergence from the previous post.)

A function is called *odd* if for all in the domain of and is called *even* if for all in the domain of . Examples for odd functions are . Examples for even functions are

**Exercise** Let be odd and be even. Assuming that the functions below are well defined, state which ones are odd and which ones are even.

- Which function is both odd and even?

(*Hint* See here for answers.)

**The Fourier coefficients of odd and even functions**

We can now use the fact that the if is odd then and that if is even, then .

- If is odd, then for all and

**Exercise*** State an analoguous result if is even. *

As we have seen above, recognising whether the function is odd or even helps to reduce the amount of calculations necessary.

**General periodic functions**

*Assume now that the function is periodic, but the period is rather than , that is *

* Then we can still define a Fourier series.*

**Question** If is periodic with period , what is the period length of the function ?

* Exercise* Show that

* *

*Let be periodic with period , bounded and integrable on . Let *

* Then the formal series *

* is called the Fourier series of .*

* Exercise* Let have period where

* Calculate the Fourier series and compare your result with the solution from here. Further, even though is not odd we still have for . How can you obtain this result without doing any integration? *

**Fourier sine and cosine series**

If a function is only defined on, say, , then we can extend it to by making either odd by setting for or even by setting for . Hence we can then calculate the Fourier series of . If we make odd, then the Fourier series is called *Fourier sine series*, whereas if we extend even, then the Fourier series is called *Fourier cosine series* (in the first case , whereas in the second case ).

* Exercise* Let

* Let be the even extension -periodic extension of and let be the odd -periodic extension of .
*

*Sketch the graphs of and on the interval .**Find and sketch its graph on .**Find and sketch its graph on .**Show that .*

Should the integrals of the cosine squared and sine squared functions both equal L, rather than 1/L?

Yes, you are right. I corrected it already. Thanks.

This is probably too late, but for odd and even functions, you stated that both f(x)cos(kx) and f(x)sin(kx) are both odd functions. f(x)sin(kx) is meant to be even, yes?

You are right. I corrected it.