Even and Odd Functions

De tion. Saying that f is an even function means that f ( x) = f (x ) for all x in th e dnoim ain of f : Saying that f is an o dd function means that f ( x) = f (x) o r f (x) = f ( x ) for all x in the domain of f . Note. The graph of an even function is symmetric about the y -axis. y 25 20 15 10 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x An Even Function The graph of an odd function is symmetric about the origin. (x; y) is on the graph if and only if ( x; y)is on the graph. y 120 100 80 60 40 20 -5 -4 -3 -2 -1 1 -20 -40 -60 -80 -100 -120 An Odd Function 1 2 3 4 5 x Note. If f (x) = xn then f is an even function when n is an even integer and f is an odd function when f is an odd integer. The cosine function is even and the sine function is odd. Theorem. Suppose that each of f and g is an even function and each of u and v is an odd function all with the same domain D. 1. f + g is an even function. 2. u + v is an odd function (unlike with integers). 3. f g is an even function. 4. u v is an even function (unlike with integers). 5. f u is an odd function (unlike with integers). Proof of (5). (f u)( x) = f ( x)u( x) = f (x) ( u(x)) = f (x)u(x) = (f u)(x) for all x in D. Suggested Problem. Prove Parts (1) - (4). Note. Most functions are neither even nor odd. For example, if f (x) = x + x2 then f ( 1) = 0 while f (1) = 2: Of course, 0 6= 2 and 0 6= 2: So f is neither even nor odd. However we do have the following fact. Theorem. If the domain of f is symmetric about 0 (meaning x is in the domain if and only if x is in the domain) then f is the sum of an even function and an odd function. Proof. Let 1 1 fe (x) = [f (x) + f ( x)] and fo (x) = [f (x) 2 2 2 f ( x)] Then 1 1 fe ( x) = [f ( x) + f ( ( x))] = [f (x) + f ( x)] = fe (x) 2 2 so fe is even; and 1 fo ( x) = [f ( x) 2 f ( ( x))] = 1 [f (x) 2 f ( x)] = fo (x) so fo is odd. Clearly f (x) = fe (x) + fo (x) De…nition. fe is called the even part of f and fo is called the odd part of f Theorem. If f is both even and odd, then f is the zero function on its domain. Proof. f ( x) = f (x) and f ( x) = Thus 2f (x) = 0 implying f (x) = 0: f (x) so f (x) = f (x) for all x in the domain of f: There is only one way to express a function as the sum of an even function and an odd function. Theorem. Suppose that f is a function whose domain is symmetric about 0: If f (x) = u1 (x) + u2 (x) = v1 (x) + v2 (x) for all x in the domain of f , each of u1 and v1 is even, and each of u2 and v2 is odd then u1 (x) = v1 (x) and u2 (x) = v2 (x) for all x in the domain of f . Proof. If f (x) = u1 (x) + u2 (x) = v1 (x) + v2 (x) then u1 (x) v1 (x) = v2 (x) u2 (x): The left side of the last equation is even and the right side is odd so each side is both even and odd. This implies that each side is 0. Thus u1 (x) = v1 (x) and v2 (x) = u2 (x) 3 Note. From Calculus, we have Z Z a f (x)dx = Z h(b) f (x)dx = Z b f (h(x))h0 (x)dx: a h(a) Theorem. If f is an even function, then Z Z L f (x)dx = 2 L L 0 Z L f (x)dx: 0 Proof. Let h(x) = x:Then Z Z 0 Z L f (x)dx + f (x)dx = L f (x)dx a b and b L f (x)dx = 0 Z 0 Z h(0) f (x)dx + h(L) L Z L f (x)dx 0 f (x)dx f (h(x))h (x)dx + 0 L Z L Z 0 f (x)dx f ( x)( 1)dx + = 0 L Z L Z L Z L Z 0 f (x)dx f (x)dx + f (x)dx = f (x)dx + = 0 0 0 L Z L f (x)dx = 2 = 0 Theorem. If f is an odd function, then Z L f (x)dx = 0: L Proof. Let h(x) = x: Z Z L f (x)dx = L = Z 0 f (x)dx + L 0 Z L f (x)dx = 0 Z 0 Z h(0) f (x)dx + h(L) L Z L 0 0 = 0 4 f (x)d 0 f (h(x))h (x)dx + f (x)dx 0 Z L = f ( x)( 1)dx + f (x)dx L 0 Z 0 Z L = ( f (x))dx + f (x)dx L 0 Z 0 Z L Z L Z = f (x)dx + f (x)dx = f (x)dx + L Z 0 L 0 L f (x)dx Theorem. If f is an even function that is integrable over [ L; L], the Fourier Series for f is fSn g where n X k x Sn (x) = A0 + Ak cos L k=1 in which Z 1 A0 = L and 2 Ak = L for k = 1; 2; : : :. Z L f (x)dx 0 L f (x) cos 0 k x dx L Proof. According to the de…nition of a Fourier Series, Sn (x) = A0 + n X k x k x + Bk sin L L Ak cos k=1 where 1 A0 = 2L 1 Ak = L Bk = Z L f (x) cos L 1 L Since the integrand is even Z f (x) sin L Z L 0 L f (x)dx; L k x dx for k = 1; 2; : : : ; and L L 1 2 A0 = 2L Z k x dx for k = 1; 2; : : : : L 1 f (x)dx = L Z L f (x)dx 0 and 1 Ak = L Z L k x 1 f (x) cos dx = 2 L L L Z L 0 k x 2 f (x) cos dx = L L Z 0 L f (x) cos k x dx: L Since the integrand is odd (the product of an even function and an odd function is an odd function), Z 1 L k x Bk = f (x) sin dx = 0 L L L Example. If f (x) = jxj for fSn g where L x L;since f is even, the Fourier Series for f is given by Sn (x) = A0 + n X k=1 5 Ak cos k x L in which 1 A0 = L and Ak = = = = = Z L 0 1 jxjdx = L Z L xdx = 0 L 2 Z Z k x 2 L k x 2 L f (x) cos dx = x cos dx L 0 L L 0 L Z L 2 L k x x=L L k x [[x sin ]x=0 1 sin dx] L k L k L 0 L L k x x=L 2 [0 + [cos ] ] L k k L x=0 2L (cos k cos 0) k2 2 2L (( 1)k 1) k2 2 for k = 1; 2; : : :. So the Fourier Series is fSn g where L 2L X ( 1)k Sn (x) = + 2 2 k2 k=1 n 1 cos k x : L Theorem. If f is an odd function that is integrable over [ L; L], the Fourier Series for f is fSn g where n X k x Sn (x) = Bk sin L k=1 in which 2 Bk = L for k = 1; 2; : : :. Z L f (x) sin 0 k x dx L Proof. According to the de…nition of a Fourier Series, Sn (x) = A0 + n X Ak cos k=1 where 1 A0 = 2L 1 Ak = L Bk = Z 1 L L f (x) cos L Z L f (x)dx; L k x dx for k = 1; 2; : : : ; and L L f (x) sin L Z k x k x + Bk sin L L k x dx for k = 1; 2; : : : : L 6 Since the integrand is odd, A0 = 0 and Ak = 0 for k = 1; 2; : : :.Since the integrand is even, Z k x 1 L f (x) sin dx Bk = L L L Z Z 2 L k x 2 L k x = f (x) sin dx = x sin dx L 0 L L 0 L Z L L 2 L k x x=L k x 1 = [[x ( ) cos ]x=0 + cos dx] L k L k L 0 2 L2 L2 k x x=L = [ ( 1)k + 2 2 [sin ] ] L k k L x=0 2L ( 1)k+1 k So the Fourier Series is fSn g where 2L X ( 1)k k x sin : k L k=1 n Sn (x) = 7