Fourier Series 

What
is a Fourier series? 

The formal way of putting it: The function space can be described by
a linear combination of orthogonal functions, called a basis of that space.
When developing a function f(x) into a Taylor series, we use a base consisting
of (1, x, x^2, x^3, ..., x^n) which spans the entire function space. In
Fourier series, we simply use a different basis, a different set of orthogonal
functions to span our space. These functions are: (1, cosx, sinx, cos2x,
sin2x, ..., cosnx, sinnx). In this task, we will construct a function which, under most simple conditions,
should be able to find the complete analytical expression of the Fourier series
of a function. This is a very basic progarm, intended for educational purposes
only. It will find a0, an and bn if for f(x) in the range a<x<b: And we need to calculate:
Below: (a) for example, is a 5element Fourier series for the function f(x)=x, while (b) is a 20element Fourier series of f(x)=x. The peak at each end is called the Gibbs Phenomenon, and is typical of Fourier series. (a) (b) 

Our
task 

We will creat a function that recieves: f(x), x, a, b and returns: a0, an,
bn and the complete sum to the Home screen. Then we will use it to find the
Fourier series of f(x)=x, as in the examples above, and demostrate the graphical
results obtained. 

Step
no. 1: Creating a new function 

We start by creating a new function, which we will call fourier. Press APPS, and open a new program/function dialog box. In the Variable field, write "fourier". Don't forget to change the type of variable from "program" to "function"  
? Oops... I forgot to make the Type a "function". Ans
No problem, though this will be a little annoying to fix. Either quit
the editor and go back to the Home screen; there delete the function by
either typing "delvar fourier" or delete it through the VarLink
screen; then dothe whole creation of the function again. 

This is the program/funciton editor. Here we start by defining the function's input parameters. It will receive an expression f, which can be a function of any variable. 

? How can that be? We have to know which variable because we need to put it in the integration expressions, don't we? Ans Yes, of course we do. But here's the catch: Suppose we let the user choose whichever variable he wants to put in f, but he must then specify that variable in the second parameter, like this: fourier(t^2, t, ..., ...). Now if we catch this second parameter in a variable, say, x, and use x throughout the function  when the function will run it will see t everywhere instead of x, because x evaluates to t! 

Step
no. 2: Writing the fourier function 

Back to our task, we have entered the four parameters this function receives: 

Next we should write down a short help line for the user. This is always a good thing to have. Follow the screenshot below:  
I have chosen to use three local variables in this function: a0x, anx, and bnx.  
? Why do you say "local" variables, and not just "variables"? Ans
Because there're important differences between the two: The last, and a very important point is that functions can use only
local variables*. This is because unlike a program, a function cannot
(among other things) write to the memory (i.e. create or modify a variable).


Create these three local variables
now, by first writing the instruction "Local", and then their
names. (You can choose "Local" from Var menu (F4) and item 3: Local. 

Now we must write down the integrals as above. This is a long but simple step. Follow the next few screenshots: 

? Ok. I understand what you're doing, but what's that funny sign before "n1" there? Maybe you've got a virus on your TI?? Ans : ) No, I don't have any virus. I hope there don't exist any for these calculators yet either. That special sign is the @ sign (in case you didn't recognise it), and it is in fact an inseprarable part of the expression @n#, which has this meaning: an arbitrary integer. (Where # is 1..255) There also exists the special expression @# (where again # is 1..255), which means: any arbitrary number (integer or not). These special expressions are used by some functions on the TI89/92+. For example, the desolve function uses @# to indicate unknown constants, which are not necessarily integer. When solving trigonometric equations, the CAS usually adds @n# multiplies of Pi, to show there is an infinite number of branches belonging to the same solution. You can get the @ sign by pressing Diamond+STO on the TI89, or 2nd+R on the TI92+. 

Now we write the last line, which returns the result to the Home screen.
Follow the next screenshot. The declaration 

That's it! You don't need to save or anything, because everything you write
is automatically saved in the function. Press 

Step
no. 3: Plotting the original function and its Fourier series 

Finally, we would like to see how this approximation fits the original function.
To obtain the series of f(x) up to, say, the 5th element, we simply type: 

The
resulting expression is a function of x, so we can immediately copy and
paste it into the Y= Editor. Use the UP arrow key to highlight the result and Diamond+Shift on the TI89, or Diamond+C on the TI92+ to copy it. Then Press Diamond+F1 (or Diamond+W on the TI92+) to go to the Y= Editor. 

(Please clear all other functions) On y1(x) we put f(x)=x : 

On y2(x) paste the expression (Diamond+ESC on the TI89, or Diamond+V on the TI92+): 

Before continuing, change the Fourier series's plotting to be with a thick line: 

Go to the Window settings and set the following settings : 

Finally, we can see the result. Go to the Graph screen now. 

Congratulations! You have completed this task!  
_____________________________________________________________________________________ Created by Andrew Cacovean, Nov. 17, 2001 
