Mathematical Methods: Fourier Series. Fourier Series: The Basics

Transcription

1 1 Mathematical Methods: Fourier Series Fourier Series: The Basics Fourier series are a method of representing periodic functions. It is a very useful and powerful tool in many situations. It is sufficiently useful that when some non-periodic problems arise transformations are used to make such problems periodic so the Fourier series can be used.

2 2 A function with period means that f(x) = f(x + ) for all x. Some well known periodic functions are listed below: Function eriod Function eriod Note: If a function is periodic with period then it also has periods 2, 3, 4, and so on. If we set 2π/a =, or a = 2π/, we see from the above table that cos(2πx/ ) and sin(2πx/ ) both have period. Similarly, cos(2nπx/ ) and sin(2nπx/ ), n = 1, 2, 3,..., both have period /n and hence also have period.

3 2 A function with period means that f(x) = f(x + ) for all x. Some well known periodic functions are listed below: Function eriod Function eriod sin x 2π cos x 2π sec x 2π tan x π Note: If a function is periodic with period then it also has periods 2, 3, 4, and so on. If we set 2π/a =, or a = 2π/, we see from the above table that cos(2πx/ ) and sin(2πx/ ) both have period. Similarly, cos(2nπx/ ) and sin(2nπx/ ), n = 1, 2, 3,..., both have period /n and hence also have period.

4 2 A function with period means that f(x) = f(x + ) for all x. Some well known periodic functions are listed below: Function eriod Function eriod sin x 2π sin ax 2π/a cos x 2π cos ax 2π/a sec x 2π tan x π Note: If a function is periodic with period then it also has periods 2, 3, 4, and so on. If we set 2π/a =, or a = 2π/, we see from the above table that cos(2πx/ ) and sin(2πx/ ) both have period. Similarly, cos(2nπx/ ) and sin(2nπx/ ), n = 1, 2, 3,..., both have period /n and hence also have period.

5 3 The essential idea behind Fourier series is to represent periodic functions in terms of a sum of well known periodic functions. Sines and cosines are chosen as they are smooth. If f(x) has period we write f(x) = a + n=1 a n cos 2nπx + b n sin 2nπx. Some people use a convention where in stead of a they have a /2. Always check to see which convetion an author is using. This leaves the problem of how to find a, a n and b n for a given function...

6 4 To find a we integrate f(x) over any interval of length, say from to : ( ) f(x) dx = a + a n cos 2nπx + b n sin 2nπx dx n=1

7 4 To find a we integrate f(x) over any interval of length, say from to : ( ) f(x) dx = a + a n cos 2nπx + b n sin 2nπx dx n=1 Assuming you can swap the order of integration and the summations we get f(x) dx = + n=1 ( a n a dx cos 2nπx dx + b n sin 2nπx dx ).

8 5 But and giving cos 2nπx dx = sin 2nπx dx = [ 2nπ [ 2nπ ] 2nπx sin ] 2nπx cos = 2nπ = 2nπ (sin 2nπ sin ) =, (cos 2nπ cos ) =, f(x) dx = a or a = 1 f(x) dx. (1) So a is just the average value of f(x).

9 6 To find all the other a n and b n we need to use the trigonometrical relations cos A cos B = 1 2 (cos(a + B) + cos(a B)), sin A sin B = 1 2 (cos(a B) cos(a + B)), sin A cos B = 1 (sin(a + B) + sin(a B)). 2

10 7 To find a n consider = cos 2mπx f(x) dx cos 2mπx ( a + n=1 a n cos 2nπx ) + b n sin 2nπx dx.

11 8 Again, assuming we can swap the order of integration and summation, we obtain cos 2mπx f(x) dx = a cos 2mπx dx + n=1 a n cos 2mπx 2nπx cos dx + b n cos 2mπx 2nπx sin dx,

12 9 + n=1 = a a n 2 cos 2mπx cos dx 2(m + n)πx + cos 2(m n)πx 2(m + n)πx 2(m n)πx + b n sin sin dx. Since m and n are both positive integers we have seen already that all these integrals are zero except for the cases where m = n. In the case m = n the sine integral is obviously, but cos(2(m n)πx/ ) = 1 and so that integral gives. Hence dx cos 2mπx f(x) dx = a m 2,

13 1 or Similarly we find a n = 2 cos 2nπx f(x) dx. (2) b n = 2 sin 2nπx f(x) dx. (3) The integrals for a, a n and b n given above are all over the interval from to. However as all the functions involved are periodic with period they can be taken over any interval of length. You are free to choose the interval to make the calculations involved easier for youself.

14 Equations (1), (2) and (3) are called the Euler formulas for the Fourier coefficients. 11

15 11 Equations (1), (2) and (3) are called the Euler formulas for the Fourier coefficients. Example: Sketch the periodic function with period 2π given by f(x) = { 1 π < x +1 < x π Find its Fourier series.