3 Fourier series 3.1 Fourier series The phases exp(inx)/p2⇡ for integer n are orthonormal on an interval of length 2⇡ 2⇡ imx inx 2⇡ i(n m)x e− e e − 1ifm = n dx = dx = δm,n = (3.1) p2⇡ p2⇡ 2⇡ 0ifm = n Z0 Z0 ⇢ 6 in which δn,m is Kronecker’s delta (1.38). So if a function f(x) is a sum of these phases, called a Fourier series, 1 einx f(x)= f , (3.2) n p n= 2⇡ X1 then the orthonormality (3.1) of these phases exp(inx)/p2⇡ gives the nth coefficient fn as the integral 2⇡ inx 2⇡ inx imx e− e− 1 e 1 f(x) dx = fm dx = δn,mfm = fn. p2⇡ p2⇡ p2⇡ Z0 Z0 m= m= X1 X1 (3.3) Fourier series can represent functions f(x) that are square integrable on the interval 0 <x<2⇡ (Joseph Fourier 1768–1830). In Dirac’s notation, we interpret the phases einx x n = (3.4) h | i p2⇡ as the components of the vector n in the x basis. These components are | i | i inner products x n of n and x . The orthonormality integral (3.1) shows h | i | i | i 102 Fourier series that the inner product of n and m is unity when n = m and zero when | i | i n = m 6 2⇡ 2⇡ ei(n m)x m n = m I n = m x x n dx = − dx = δ . (3.5) h | i h | | i h | ih | i 2⇡ m,n Z0 Z0 Here I is the identity operator of the space spanned by the vectors x | i 2⇡ I = x x dx. (3.6) | ih | Z0 Since the vectors n are orthonormal, a sum of their outer products n n | i | ih | also represents the identity operator 1 I = n n (3.7) | ih | n= X1 of the space they span, which is the same as the space spanned by the vectors x . This representation of the identity operator, together with the formula | i (3.4) for x n , shows that the inner product f(x)= x f ,whichisthe h | i h | i component of the vector f in the x basis, is given by the Fourier series | i | i (3.2) 1 f(x)= x f = x I f = x n n f h | i h | | i h | ih | i n= X1 (3.8) 1 einx 1 einx = n f = f . p h | i p n n= 2⇡ n= 2⇡ X1 X1 Similarly, the other representation (3.6) of the identity operator shows that the inner products f = n f , which are the components of the vector f n h | i | i in the n basis, are the Fourier integrals (3.3) | i 2⇡ 2⇡ inx e− fn = n f = n I f = n x x f dx = f(x) dx. (3.9) h | i h | | i h | ih | i p2⇡ Z0 Z0 The two representations (3.6 & 3.7) of the identity operator also give two ways of writing the inner product g f of two vectors f and g h | i | i | i 1 1 g f = g n n f = g⇤f h | i h | ih | i n n n= n= 1 1 (3.10) X2⇡ X2⇡ = g x x f dx = g⇤(x) f(x) dx. h | ih | i Z0 Z0 When the vectors are the same, this identity shows that the sum of the 3.1 Fourier series 103 squared absolute values of the Fourier coefficients fn is equal to the integral of the squared absolute value f(x) 2 | | 1 1 2⇡ 2⇡ f f = n f 2 = f 2 = x f 2 dx = f(x) 2 dx. h | i |h | i| | n| |h | i| | | n= n= 0 0 X1 X1 Z Z (3.11) The Fourier series (3.2 & 3.8) are periodic with period 2⇡ because the phases x n are periodic with period 2⇡,exp(in(x +2⇡)) = exp(inx). Thus h | i even if the function f(x) which we use in (3.3 & 3.9) to make the Fourier coefficients f = n f is not periodic, its Fourier series (3.2 & 3.8) will n h | i nevertheless be strictly periodic, as illustrated by Figs. 3.2 & 3.4. The complex conjugate of the Fourier series (3.2 & 3.8) is inx inx 1 e− 1 e f ⇤(x)= fn⇤ = f ⇤ n (3.12) p − p n= 2⇡ n= 2⇡ X1 X1 so the nth Fourier coefficient fn(f ⇤) for f ⇤(x) is the complex conjugate of the nth Fourier coefficient for f(x) − fn(f ⇤)=f ⇤ n(f). (3.13) − Thus if the function f(x) is real, then fn(f)=fn(f ⇤)=f ⇤ n(f) or fn = f ⇤ n. (3.14) − − Example 3.1 (Fourier Series by Inspection) The doubly exponential func- tion exp(exp(ix)) has the Fourier series 1 1 exp eix = einx (3.15) n! n=0 X in which n!=n(n 1) ...1isn-factorial with 0! 1. − ⌘ Example 3.2 (Beats) The sum of two sines f(x)=sin!1x +sin!2x of similar frequencies ! ! is the product (exercise 3.1) 1 ⇡ 2 f(x) = 2 cos 1 (! ! )x sin 1 (! + ! )x. (3.16) 2 1 − 2 2 1 2 1 The first factor cos 2 (!1 !2)x is the beat; it modulates the second factor 1 − sin 2 (!1 + !2)x as illustrated by Fig. 3.1. 104 Fourier series Beats 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0 1 2 3 4 5 6 7 8 9 10 Figure 3.1 The curve sin !1x +sin!2x for !1 = 30 and !2 = 32. Mat- lab scripts for this chapter’s figures are in Fourier series at github.com/ kevinecahill. Example 3.3 (Laplace’s equation) The Fourier series (exercise 3.2) 1 n 2⇡ in✓0 in✓ ⇢ | | e− e f(⇢,✓)= h(✓0) d✓0 (3.17) a p p n= " 0 2⇡ # 2⇡ X1 ⇣ ⌘ Z (Ritt, 1970, p. 3) obeys Laplace’s equation (7.23) 1 d df 1 @2f ⇢ + = 0 (3.18) ⇢ d⇢ d⇢ ⇢2 @✓2 ✓ ◆ for ⇢<aand respects the boundary condition f(a, ✓)=h(✓). 3.2 The interval In section 3.1, we singled out the interval [0, 2⇡], but to represent a periodic function f(x) of period 2⇡, we could have used any interval of length 2⇡, such as the interval [ ⇡,⇡] or [r, r +2⇡] − r+2⇡ inx dx fn = e− f(x) . (3.19) p2⇡ Zr 3.3 Where to Put the 2pi’s 105 This integral is independent of its lower limit r when the function f(x)is periodic with period 2⇡. The choice r = ⇡ is often convenient. With this − choice of interval, the coefficient f is the integral (3.3) shifted by ⇡ n − ⇡ inx dx fn = e− f(x) . (3.20) ⇡ p2⇡ Z− But if the function f(x) is not periodic with period 2⇡, then the Fourier coefficients (3.19) do depend upon the choice r of interval. 3.3 Where to Put the 2pi’s In sections 3.1–3.2, we used the orthonormal functions exp(inx)/p2⇡, and so we had factors of 1/p2⇡ in the Fourier equations (3.2, 3.3, 3.8 , and 3.9). One can avoid these square roots by setting dn = fn/p2⇡ and writing the Fourier series (3.2) and the orthonormality relation (3.3) as 2⇡ 1 inx 1 inx f(x)= d e and d = dx e− f(x) (3.21) n n 2⇡ n= 0 X1 Z or by setting cn = p2⇡fn and using the rules ⇡ 1 1 inx inx f(x)= c e and c = f(x) e− dx. (3.22) 2⇡ n n n= ⇡ X1 Z− The cost of these asymmetrical notations is that factors of 2⇡ pop up (exer- cise 3.3) in equations (3.10 & 3.11) for the inner products g f and f f . h | i h | i Example 3.4 (Fourier Series for exp( m x )) Let’s compute the Fourier − | | series for the real function f(x)=exp( m x ) on the interval ( ⇡,⇡). Using − | | − the shifted interval (3.20) and the 2⇡-placement convention (3.21), we find that the coefficient dn is the integral ⇡ dx inx m x dn = e− e− | | (3.23) ⇡ 2⇡ Z− which we may do as two simpler integrals 0 ⇡ dx (m in)x dx (m+in)x dn = e − + e− ⇡ 2⇡ 0 2⇡ Z− Z (3.24) 1 m n ⇡m = 1 ( 1) e− ⇡ m2 + n2 − − ⇥ ⇤ which shows that dn = d n.Sincem is real, the coefficients dn also are − 106 Fourier series 2 x Fourier series for e− | | 1 0.8 0.6 0.4 0.2 0 -6 -4 -2 0 2 4 6 Figure 3.2 The 10-term (dashes) Fourier series (3.25) for the function exp( 2 x ) on the interval ( ⇡,⇡) is plotted from 2⇡ to 2⇡. All Fourier series− are| | periodic, but the function− exp( 2 x ) (solid)− is not. − | | real, dn = dn⇤ . They therefore satisfy the condition (3.14) that holds for real functions, dn = d⇤ n, and give the Fourier series for exp( m x ) as − − | | m x 1 inx 1 1 m n ⇡m inx e− | | = d e = 1 ( 1) e− e n ⇡ m2 + n2 − − n= n= X1 X1 (3.25) ⇡m ⇥ ⇤ (1 e− ) 1 2 m n ⇡m = − + 1 ( 1) e− cos(nx). m⇡ ⇡ m2 + n2 − − n=1 X ⇥ ⇤ In Fig. 3.2, the 10-term (dashes) Fourier series for m = 2 is plotted from x = 2⇡ to x =2⇡.