Periodic Functions And Fourier Series
Periodic Functions And Fourier Series :- Definition : A function f is said to be periodic with period T if (i) f(x) is defined for all x and f(x+T) = f(x) for all x for some positive number T . For example , sin x is periodic with period 2π , since sin (x+2π) = sin x . A periodic function has many periods, for if f(x) = f(x+T) then f(x) = f(x+T) = f(x+2T) = ....=f(x+nT), where n is any integer. Hence when T is a period of f, nT is also a period of f , but while referring to period we mean the smallest one . Since each of the functions sin x , cos x , sin 2x , cos 2x , .... are of period 2π , we may think of representing a given function f of period 2π by an infinite series of these function as ∞ f(x) = a₀/2 + Σ (aₙ cos nx + bₙ sin nx) ......(1) n=1 Now two questions arise : (i) Supposing the representation (1) is possible , how are the aₙ 's