FOURIER SERIES PROF. SEBASTIAN VATTAMATTAM
The Fourier series is named after the French scientist and mathematician Joseph Fourier(1768 - 1830), who used them in his work on heat conduction. Areas of its application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. In some of the problems below, you are asked to derive a result : π/4 = 1 − 1/3 + 1/5 − 1/7 + .... This is usually referred to as Gregory’e Series. James Gregory(1638 - 1675), a Scottish Mathematician, proved it in 1671. But, you might be surprised to know that the same series with proof is contained in a Malayalam book called Yuktibhasha witten in AD 1530. In fact, the series was derived by a Malayalee mathematician, Madhava of Sangamagrama, in the 14th century, i.e. three centuries before James Gregory. This result was used by Madhava to calculate approximate values of π. Today it is often referred to, at least by non-Indians, as Madhava-Gregory series. 1. FOURIER SERIES - General Formula The expansion is based on Dirichlet Conditions named after Johann Peter Gustav Dirichlet (1805 - 1859). Derived from the French ‘De Richelet’, Dirichlet is pronounced ‘dirishl´e. Definition 1.1. Dirichlet Conditions Let a, b ∈ R, a ≤ b, f : [a, b] → R. The function f is said to satisfy the Dirichlet Conditions in the interval [a, b] if (1) f is periodic, (2) f has a finite number of discontinuities in each period, and, (3) f has at the most a finite number of maxima and minima. Theorem 1.2. If f satisfies the Dirichlet conditions over the interval [a, b], then for a point x ∈ [a, b] of continuity, P∞ 2πnx + b sin f (x) = a20 + n=1 an cos 2πnx n b−a b−a , Rb 2 2πnx where an = b−a a f (x) cos b−a Rb 2 and bn = b−a f (x) sin 2πnx b−a a This formula being applicable to all the different cases, it is advisable to remember this formula, rather than learning the individual formula for each case. Because of this convenience we don’t treat the individual cases separately. Theorem 1.3. The Fourier series converges to 12 (f (x + 0) + f (x − 0)) if x is a point of discontinuity in (a, b) and to 21 (f (a + 0) + f (b − 0)) if x = a or x = b. 2. Expansion of even or odd functions Definition 2.1. A function f : [−L, L] → R is even if f (−x) = f (x), ∀x ∈ [−L, L] and it is odd if f (−x) = −f (x), ∀x ∈ [−L, L]. 1