2M2. Fourier Series Prof Bill Lionheart

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1 M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier coefficients of the function f(x). A function f is periodic with period if f(x + ) = f(x) for a x, or specified just on a segment of ength, such as [, ] or [0, ]. In the atter case f(x) can be extended to a periodic function of period. Given a periodic function f(x) (with period ), its Fourier coefficients a n, b n can be found by the foowing formuae: a 0 = 1 f(x) dx, and for n > 0. a n = 1 b n = 1 f(x) cos πnx f(x) sin πnx dx, dx. A function is caed even if f( x) = f(x) for a x. Simiary, a function is caed odd if f( x) = f(x) for a x. Exampes: f(x) = x is odd (it is not a periodic function!); f(x) = x is even (again not periodic); f(x) = cos x is even; f(x) = sin x is odd. This makes sense ony for functions defined on the whoe rea ine. If a function is defined on a segment [0, ], it can be extended to a periodic even function or to a periodic odd function (both with period ), by setting f( x) = f(x) or f( x) = f(x) respectivey, for < x < 0, and then extending by periodicity. If f(x) is an even periodic function, then its Fourier expansion contains ony cosines (a coefficients b n are zero). If f(x) is an odd periodic function, 1

2 then its Fourier expansion contains ony sines (a coefficients a n, incuding a 0, are zero). 3. Dirichet Theorem. Suppose a periodic function f(x) within one period (i.e., on an arbitrary segment of ength equa to the period) has ony a finite number of finite jumps, is continuous at a other points and has ony a finite number of maxima and minima. Then its Fourier series converges either to f(x) if f is continuous at x or to the midde vaue 1 ( f(x 0)) + f(x + 0) ) if there is a jump. Here f(x 0) and f(x + 0) denote eft and right imits, respectivey. 4. Fourier series are best understood in the foowing geometrica anguage. Periodic functions f(x) (with a fixed period ) are considered as anaogs of vectors. The Fourier expansion of a function f(x) is considered as an anaog of the expansion a = a 1 i + a j + a 3 k of a vector a. The scaar product (or inner product) of two functions is defined as (f, g) = f(x)g(x) dx. In fact, the integra can be taken over any segment of ength, e.g., over [0, ]. It is the anaog of the dot product of vectors: a b, which is often denoted aso as (a, b). The foowing reations for the system of functions 1, cos πx, sin πx, cos πx, sin πx,... are caed orthogonaity: cos πnx sin πnx cos πmx dx = 0 cos πnx sin πmx sin πmx dx = 0 if n m dx = 0 if n m for a n and m. They are anaogous to the fact that i, j, k are mutuay perpendicuar. Arbitrary two functions f and g are caed orthogona if their scaar product vanishes: (f, g) = 0. The norm of a function f(x) is the

3 square root of its scaar square : f = (f, f) = (f(x)) dx. It is the anaog of the magnitude or ength of a vector. Exampe. One can easiy find that and that cos πnx sin πnx 1 = (1, 1) = = (cos πnx = (sin πnx dx =, cos πnx ) =, sin πnx ) = cos πnx dx = sin πnx dx = (for any number n). One shoud compare it with i, j, k being unit vectors, i.e., of norm or magnitude 1. The above formuae for the Fourier coefficients foow from here and the orthogonaity reations. Parseva Theorem. For a -periodic function f(x) its norm can be expressed via the Fourier coefficients as foows: f = a 0 + ( ) a n + b n. It is the anaog of the Pythagoras theorem for vectors: a = a 1+a +a 3 where a = a 1 i + a j + a 3 k. The appearance of extra coefficients such as or is expained by the basis eements being not unit (compared with i, j, k). Fourier expansions can be best understood in the sense of convergence in mean : for any function f(x) with integrabe square, its Fourier series converges to f(x) in the sense of the above-defined norm: f S N 0 when N 3

4 where S N is the partia Fourier sum for f(x): S N (x) = a 0 + N ( a n cos πnx + b n sin πnx ). 5. A the above was for functions taking vaues in rea numbers. It aso hods for compex-vaued functions, but the definition of the scaar product shoud be modified by incuding compex conjugation: In particuar (f, g) = f = (f, f) = f(x)g(x) dx. f(x) dx. Reca the Euer formua connecting trigonometric functions and exponentias: e iα = cos α + i sin α, from where we have cos α = 1 (eiα + e iα ), sin α = 1 i (eiα e iα ). Ceary, a Fourier series can be rewritten in terms of compex exponentias. For a given -periodic function f(x), rea or compex, its Fourier series (or Fourier expansion) in the compex form has the appearance: f(x) = + n= c n e iπnx. (Compared with it, the Fourier series in sines and cosines for a rea function f(x) wi be referred to as the Fourier series in the rea form.) The coefficients c n, where n = 0, ±1, ±,..., can be found by the formuae c n = 1 f(x)e iπnx This foows from the orthogonaity: (e iπnx, e iπmx ) = e iπnx dx = 1 (f, e iπnx ). e iπmx dx = 0 for n m 4

5 (proof is very simpe) and the equaity e iπnx = e iπnx e iπnx dx = for a n (the orthogonaity reations for sines and cosines immediatey foow from the orthogonaity of exponentias). Parseva Theorem in the compex form. For a -periodic rea or compex function f(x) its norm can be expressed via the Fourier coefficients c n as foows: f = + n= c n. Notice that c n are in genera compex, so c n = c n c n. Using Euer s formua, the Fourier expansion in the compex form can be expressed in terms of sines and cosines, and back. If the function f(x) is rea, then the coefficients of its Fourier expansion in the compex form satisfy the extra condition c n = c n for a n, and they are reated with the coefficients of the Fourier expansion in the rea form as foows: and c 0 = a 0 c n = 1 (a n ib n ) for n > 0 c n = 1 (a n + ib n ) for n < 0, where n > 0. a 0 = c 0 a n = c n + c n b n = i(c n c n ) 5