Fourier Series: Deﬁnitions and Coefﬁcients We will ﬁrst state Fourier’s theorem for periodic functions with period P = 2π. In words, the theorem says that a function with period 2π can be written as a sum of cosines and sines which all have period 2π. Theorem (Fourier) Suppose f (t) has period 2π then we have f (t) ∼ = a0 + a1 cos(t) + a2 cos(2t) + a3 cos(3t) + . . . 2 +b1 sin(t) + b2 sin(2t) + b3 sin(3t) + . . . ∞ a0 + ∑ an cos(nt) + bn sin(nt), 2 n =1 (1) where the coefﬁcients a0 , a1 , . . . and b1 , b2 . . . are computed by 1 π a0 = f (t) dt π −π � π 1 an = f (t) cos(nt) dt π −π � 1 π bn = f (t) sin(nt) dt π −π � (2) Some comments are in order. 1. As we saw in the quiz above, each of the functions cos(t), cos(2t), cos(3t), . . . all have 2π as a period. The same is clearly true for sin(t), sin(2t), sin(3t), . . . . 2. The series on the right-hand side (1) is called a Fourier series; and the coefﬁcients , a0 , a1 , . . . b1 , b2 , . . . in (2) are called the Fourier coefﬁcients of f (t). 3. The letter a is used in a0 /2 because we can think of it as the coefﬁcient of cos(0 · t) = 1. We don’t need a b0 term because sin(0 · t) = 0. The term constant term a20 is written in this way to make the formula for a0 look just like those of the other cosine coefﬁcients an . (We will see why we need the factor of 12 in a later note when we prove that these formulas really do give the coefﬁcients.) 4. In (1) we used the symbol ∼ instead of an equal sign because the two sides of (1) might differ at those values of t where f (t) is discontinuous. For us, this is a minor point and we will allow ourselves to use an equal sign from now on. 18.03SC 5. There is some terminology coming from acoustics and music: the n = 1 frequency is called the fundamental, and the frequencies n ≥ 2 are called the higher harmonics (or overtones). We will explore the con nection between Fourier series and sound in a later session. Fourier series are a wonderful tool for breaking a periodic function, however complicated, into simple pieces. The superposition principle will then allow us to solve DE’s with arbitrary periodic input in Fourier series form. In later notes we will extend Fourier’s theorem to functions of other periods. The extension is straighforward, but requires more notation, so we will wait until you have gained some experience with Fourier series.