Fourier series Dr. Kamlesh Jangid Department of HEAS (Mathematics) Rajasthan Technical University, Kota-324010, India E-mail: [email protected] Dr. Kamlesh Jangid (RTU Kota) Fourier series 1 / 17 Outline Outline 1 Fourier Series 2 Convergence and Sum of a Fourier Series Dr. Kamlesh Jangid (RTU Kota) Fourier series 2 / 17 Outline Overview Fourier series are infinite series designed to represent general periodic functions in terms of simple ones, namely, cosines and sines. They constitute a very important tool, in particular in solving problems that involve ODEs and PDEs. Fourier series are in a certain sense more universal than the familiar Taylor series in calculus because many discontinuous periodic functions of practical interest can be developed in Fourier series but, of course, do not have Taylor series representations. Dr. Kamlesh Jangid (RTU Kota) Fourier series 3 / 17 Fourier Series Fourier series Fourier series are the basic tool for representing periodic functions. A function f (x) is called a periodic function, if f (x) is defined for all real x (perhaps except at some points) and if there is some positive number p, called a period of f (x), such that f (x + p) = f (x) for all x: (1) The smallest positive period is often called the fundamental period. Dr. Kamlesh Jangid (RTU Kota) Fourier series 4 / 17 Fourier Series Familiar periodic functions are sin x, cos x, tan x, and cot x. Examples of functions that are not periodic are x; x2; x3, ex, cosh x, and ln x. If f (x) has period p, it also has the period 2p because (1) implies f (x + 2p) = f ([x + p] + p) = f (x + p) = f (x) Now, we represent a function f (x) defined in [−π; π] of period 2π in terms of the simple functions 1; cos x; sin x; cos 2x; sin 2x; ··· ; cos nx; sin nx; ··· (2) All these functions have the period 2π. Dr. Kamlesh Jangid (RTU Kota) Fourier series 5 / 17 Fourier Series Now suppose that f (x) is a given function of period 2π and is such that it can be represented by a convergent series (known as Fourier series) and, moreover, has the sum f (x). Then f (x) = a0 + a1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + ··· 1 X or f (x) = a0 + (an cos nx + bn sin nx) (3) n=1 The coefficients of (3) are the Fourier coefficients of f (x), given by the Euler formulas( for the detailed derivation go through any book on Fourier series) Dr. Kamlesh Jangid (RTU Kota) Fourier series 6 / 17 Fourier Series 9 a = 1 R π f (x) dx; > 0 2π −π > > => 1 R π (4) an = π −π f (x) cos nx dx; n = 1; 2; 3; ::: > > 1 R π > bn = π −π f (x) sin nx dx; n = 1; 2; 3; ::: ; Definition Let f (x) be periodic with period 2π and piecewise continuous in the interval [−π; π]. Then the Fourier series of f (x) is given by (3) with coefficients (4). Dr. Kamlesh Jangid (RTU Kota) Fourier series 7 / 17 Convergence and Sum of a Fourier Series Convergence and Sum of a Fourier Series Theorem (Sufficient condition) Let f (x) and f 0(x) be piecewise continuous on the interval [−π; π]. Then, the Fourier series (3) of f (x) on this interval converges to f (x) at a point of continuity. At a point x0 of discontinuity, the Fourier series converges to 1 [f (x + 0) + f (x − 0)] 2 0 0 where f (x0 + 0) and f (x0 − 0) are the right-and left-hand limits, respectively. Dr. Kamlesh Jangid (RTU Kota) Fourier series 8 / 17 Convergence and Sum of a Fourier Series Example Find the Fourier series expansion of the following periodic function with period 2π 8 < π + x; if − π < x < 0 f (x) = : 0; if 0 ≤ x < π Solution The Fourier series of f (x) is given by 1 X f (x) = a0 + (an cos nx + bn sin nx) n=1 where the Fourier coefficients are obtained as follows: Dr. Kamlesh Jangid (RTU Kota) Fourier series 9 / 17 Convergence and Sum of a Fourier Series 1 Z π 1 Z 0 π a0 = f (x)dx = (π + x)dx = : 2π −π 2π −π 4 1 Z π 1 Z 0 Z 0 an = f (x) cos(nx)dx = [ π cos(nx)dx + x cos(nx)dx] π −π π −π −π 1 sin(nx) sin nx cos nx0 = π + x + 2 π n n n −π 1 1 1 = (1 − cos nπ) = [1 − (−1)n] π n2 πn2 8 < 0; for n even = 2 : ; for n odd: πn2 Dr. Kamlesh Jangid (RTU Kota) Fourier series 10 / 17 Convergence and Sum of a Fourier Series 1 Z 0 1 − cos nx sin nx0 b = (π + x) sin(nx)dx = (π + x)( ) + n 2 π −π π n n −π 1 −π −1 = ( ) = : π n n Therefore, 1 π X 1 1 f (x) = + (1 − (−1)n) cos nx − sin nx 4 πn2 n n=1 π 2 cos x cos 3x sin x sin 2x = + + + ··· − + + ··· : 4 π 12 32 1 2 Dr. Kamlesh Jangid (RTU Kota) Fourier series 11 / 17 Convergence and Sum of a Fourier Series Fourier Series for a Function of any period p = 2` Let f (x) be periodic with period 2` and piecewise continuous in the interval [−`; `]. Then the Fourier series of f (x) is given by 1 X h nπ nπ i f (x) = a + a cos x + b sin x (5) 0 n ` n ` n=1 where 1 R ` 9 a0 = f (x) dx; > 2` −` > => 1 R ` nπ an = f (x) cos x dx; n = 1; 2; 3; ::: (6) ` −` ` > > 1 R ` nπ ;> bn = ` −` f (x) sin ` x dx; n = 1; 2; 3; ::: Dr. Kamlesh Jangid (RTU Kota) Fourier series 12 / 17 Convergence and Sum of a Fourier Series Example Find the Fourier series expansion of the following periodic function of period p = 2` = 4. 8 < 2 + x; if − 2 ≤ x ≤ 0 f (x) = : 2 − x; if 0 ≤ x ≤ 2: Solution The Fourier series of f (x) is given by 1 X h nπ nπ i f (x) = a + a cos x + b sin x 0 n 2 n 2 n=1 where the Fourier coefficients are obtained as follows: Dr. Kamlesh Jangid (RTU Kota) Fourier series 13 / 17 Convergence and Sum of a Fourier Series 1 Z 2 1 Z 0 Z 2 a0 = f (x)dx = (2 + x)dx + (2 − x)dx = 1: 4 −2 4 −2 0 1 Z 0 nπx Z 2 nπx an = (2 + x) cos( )dx + (2 − x) cos( )dx 2 −2 2 0 2 " 1 sin(nπx=2) cos(nπx=2)0 = ( + x) + 2 2 2 (nπ=2) (nπ=2) −2 # sin(nπx=2) cos(nπx=2)2 + ( − x) − 2 2 (nπ=2) (nπ=2) 0 1 1 cos(nπ) 1 cos(nπ) = − + − 2 (nπ=2)2 (nπ=2)2 (nπ=2)2 (nπ=2)2 ( 4 0; for n even = [1 − (−1)n] = n2π2 8 n2π2 ; for n odd : Dr. Kamlesh Jangid (RTU Kota) Fourier series 14 / 17 Convergence and Sum of a Fourier Series 1 Z 0 nπx Z 2 nπx bn = (2 + x) sin( )dx + (2 − x) sin( )dx = 0: 2 −2 2 0 2 (7) Therefore, the Fourier series expansion is given by 1 8 X 1 πx f (x) = 1 + cos[(2n − 1) ] π2 (2n − 1)2 2 n=1 8 1 nπ 1 3πx = 1 + cos( ) + cos( ) + ::: : π2 12 2 32 2 Dr. Kamlesh Jangid (RTU Kota) Fourier series 15 / 17 Convergence and Sum of a Fourier Series Exercise Example (i) A sinusoidal voltage E sin !t, where t is time, is passed through a half-wave rectifier that clips the negative portion of the wave. Find the Fourier series of the resulting periodic function with period p = 2` = 2π=! ( 0; −` < t < 0; u(t) = E sin !t; 0 < t < `: (ii) Find the Fourier series of the given function f (x) with period 2π ( x2; −π=2 < t < π=2 f (x) = π2=4; π=2 < t < 3π=2: Dr. Kamlesh Jangid (RTU Kota) Fourier series 16 / 17 Convergence and Sum of a Fourier Series For the video lecture use the following link https://youtube.com/channel/ UCk9ICMqdkO0GREITx-2UaEw THANK YOU Dr. Kamlesh Jangid (RTU Kota) Fourier series 17 / 17.