Even and Odd Functions If a periodic function f (t) is an even function we have already used the fact that its Fourier series will involve only cosines. Likewise the Fourier series of an odd function will contain only sines. Here we will give short proofs of these statements. Even and odd functions. Deﬁnition. A function f (t) is called even if f (−t) = f (t) for all t. The graph of an even function is symmetric about the y-axis. Here are some examples of even functions: 1. t2 , t4 , t6 , . . . , any even power of t. 2. cos( at) (recall the power series for cos( at) has only even powers of t). 3. A constant function is even. We will need the following fact about the integral of an even function over a ’balanced’ interval [− L, L]. If f (t) is even then � L −L f (t) dt = 2 � L f (t) dt. 0 This fact becomes clear if we think of the integral as an area (see ﬁg. 1). t −L L Fig. 1: Even functions: ( total area = twice area of right half) t −L L Fig. 2: Odd functions: (total (signed) area is 0) Deﬁnition. A function f (t) is called odd if f (−t) = − f (t) for all t. The graph of an odd function is symmetric about the the origin. Here are some examples of odd functions: 1. t, t3 , t5 , . . . , any odd power of t. 2. sin( at) (recall the power series for sin( at) has only odd powers of t). Even and Odd Functions OCW 18.03SC We will need the following fact about the integral of an odd function over a ’balanced’ interval [− L, L]. If f (t) is odd then � L −L f (t) dt = 0. This fact becomes clear if we think of the integral as an area (see Fig. 2). Multiplying Even and Odd Functions When multiplying even and odd functions it is helpful to think in terms of multiply even and odd powers of t. This gives the following rules. 1. even × even = even 2. odd × odd = even 3. odd × even = odd All this leads to the even and odd Fourier coefﬁcient rules: Assume f (t) is periodic then: � � π � 2 L 1. If f (t) is even then we have bn = 0, and an = f (t) cos n t dt. L 0 L � L � 2 π � 2. If f (t) is odd then we have an = 0, and bn = f (t) sin n t dt. L 0 L Reason: Assume f (t) is even. The rule for multiplying even functions tells us that f (t) cos at is even and the rule for integrating an even function over a symmetric interval tell us that an = 1 L � L −L � � π � � π � 2 L f (t) cos n t dt = f (t) cos n t dt. L L 0 L Likewise, the rule even × odd = odd tell us that f (t) sin at is odd, and so the integral for bn is 0. If f (t) is odd everything works much the same. The rule for multiplying odd functions tells us that f (t) sin at is even and therefore 1 bn = L � L −L � � π � � π � 2 L f (t) sin n t dt = f (t) sin n t dt. L L 0 L Likewise the rule odd × even = odd tells us that f (t) cos( at) is odd, and so the integral for an is 0. Examples: In previous sessions we saw the odd square wave had only sine coefﬁcients and the even triangle wave had only cosine coefﬁcients. 2