Chapter2(maths3).pdf.PDF

CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x, , where T is a positive constant. The least value of T0 is called the period of . EXAMPLES We know that = sin x = sin (x + 4 ) = … Therefore the function has period 2 , 4 , 6 , etc. However, 2 is the least value and therefore is the period of f(x). Similarly cos x is a periodic function with the period 2 and tan x has period . DIRICHLET’S CONDITIONS A function defined in c x c+2l can be expanded as an infinite trigonometric series of the form + provided 1. is single- valued and finite in (c , c+2l) 2. is continuous or piecewise continuous with finite number of finite discontinuities in (c , c+2l). 3. has no or finite number of maxima or minima in (c , c+2l). EULER’S FORMULAS If a function defined in (c , c+2l) can be expanded as the infinite trigonometric series + then [ Formulas given above for and are called Euler’s formulas for Fourier coefficients] DEFINITION OF FOURIER SERIES The infinite trigonometric series + is called the Fourier series of in the interval c x c+2l, provided the coefficients are given by the Euler’s formulas. EVEN FUNCTION If = in (-l , l) such that = , then is said to be an even function of x in (-l , l). If Such that = or = , then is said to be an even function of x in (-l , l). EXAMPLE y = cos x , y = are even functions. ODD FUNCTION If = in (-l , l) such that