DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE

MATHEMATICS-II LECTURE-12 FOURIER SERIES OF A PERIODIC FUNCTION OF PERIOD [Chapter – 10.2] TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION]

DEPARTMENT OF MATHEMATICS, CVRCE OUTLINES  Introduction  Orthogonality of the trigonometric System  Euler’s formula for the Fourier coefficients  Fourier series of a periodic function of period 2 .  Convergence and Sum of Fourier Series  Some problems based on these methods

DEPARTMENT OF MATHEMATICS, CVRCE Introduction   Fourier Series introduced in 1807 by Joseph Fourier, A French Physicist and Mathematician.  Fourier series is an infinite series representation of periodic functions in terms of the trigonometric sine and cosine function whose coefficients are determined from by the Euler’s formulae. Jean-Baptiste Joseph Fourier (March 21 st 1768-May16th 1830)

DEPARTMENT OF MATHEMATICS, CVRCE  Fourier series is very useful in the study of Heat conduction, Mechanics, Concentration of chemicals and pollutants, Electrostatics, CAT scan, etc.  Fourier series is very powerful method to solve Ordinary and Partial differential equations with periodic functions appearing as non- homogeneous terms.

DEPARTMENT OF MATHEMATICS, CVRCE Orthogonality of the Trigonometric System on the interval Let n and m be integers with n  0, m  0. Then

DEPARTMENT OF MATHEMATICS, CVRCE Let n and m be integers with n  0, m  0, n  m. Then Orthogonality of the Trigonometric System on the interval

DEPARTMENT OF MATHEMATICS, CVRCE Orthogonality of the Trigonometric System on the interval Let n and m be integers with n  0, m  0, n = m. Then

DEPARTMENT OF MATHEMATICS, CVRCE Euler Formulae for the Fourier Coefficients: Let us assume that be a periodic function of period and integrable over the interval be the sum of the trigonometric series where are known as the Fourier coefficients of and is given by the following integrals: Euler Formulae for the Fourier Coefficients: Let us assume that be a periodic function of period and integrable over the interval be the sum of the trigonometric series where are known as the Fourier coefficients of and is given by the following integrals: FOURIER SERIES OF PERIODIC FUNCTIONS WITH PERIOD 2 

DEPARTMENT OF MATHEMATICS, CVRCE Euler’s formula contd… The above representations (2) are also called the Euler’s formulae.

DEPARTMENT OF MATHEMATICS, CVRCE Determination of the coefficient term a 0 Integrating on both sides of eqn.(1) from we get

Determination of the coefficients a n of the cosine terms Multiplying eqn.(1) by is any fixed positive integer, and then integrate from we get

DEPARTMENT OF MATHEMATICS, CVRCE Determination of the coefficients a n of the cosine terms

DEPARTMENT OF MATHEMATICS, CVRCE Determination of the coefficients a n of the cosine terms Replacing m with n in the above equation we get

DEPARTMENT OF MATHEMATICS, CVRCE Determination of the coefficients b n of the cosine terms Multiplying eqn.(1) by is any fixed positive integer, and then integrate from we get

DEPARTMENT OF MATHEMATICS, CVRCE Determination of the coefficients a n of the cosine terms

DEPARTMENT OF MATHEMATICS, CVRCE Determination of the coefficients a n of the cosine terms Replacing m with n in the above equation we get

DEPARTMENT OF MATHEMATICS, CVRCE Fourier Series Fourier series of a periodic function with period is a trigonometric series which is given by the eqn.(1) with the Fourier coefficients given by the Euler’s formula (2). The individual terms in Fourier series are known as harmonic.

DEPARTMENT OF MATHEMATICS, CVRCE Representation of Fourier series of in the interval :- Representation of Fourier series of in the interval :- If we put in eqn.(2), then the interval becomes and the formula is given by the following integrals: SOME SPECIAL CASES

DEPARTMENT OF MATHEMATICS, CVRCE Representation of Fourier series of in the interval :- If we put in eqn.(2), then the interval becomes and the formula is given by the following integrals: If we put in eqn.(2), then the interval becomes and the formula is given by the following integrals:

DEPARTMENT OF MATHEMATICS, CVRCE Representation of Fourier series of in the interval Representation of Fourier series of in the interval If we put in eqn.(2), then the interval becomes and the formula is given by the following integrals: If we put in eqn.(2), then the interval becomes and the formula is given by the following integrals:

DEPARTMENT OF MATHEMATICS, CVRCE Method of obtaining Fourier series of Fourier series of the periodic function with period 2  is

DEPARTMENT OF MATHEMATICS, CVRCE Example(1): Sketch or plot and find the Fourier series of the periodic function of period 2  Example(1): Sketch or plot and find the Fourier series of the periodic function of period 2  Solution:

DEPARTMENT OF MATHEMATIC S, CVRCE The Fourier series of the given function is where

DEPARTMENT OF MATHEMATICS, CVRCE

Therefore, the required Fourier series is

DEPARTMENT OF MATHEMATICS, CVRCE Convergence and Sum of Fourier Series If a periodic function f(x) with period 2  is piecewise continuous in the interval -  ≤ x ≤  and has a left-hand derivative and right-hand derivative at each point of the interval, then the Fourier series of f(x) is convergent. Its sum is f(x), except at a point x 0 for which the function f(x) is discontinuous and the sum of the series is the average of left and right-hand limit of f(x) at x 0, i.e.,

DEPARTMENT OF MATHEMATICS, CVRCE Example(2): Find the Fourier series expansion for, if Hence deduce that Example(2): Find the Fourier series expansion for, if Hence deduce that Solution Fourier series of the given function is where

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Finally,

DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of a’s and b’s in eqn.(1), we get which is the required result.

Putting in eqn.(2), we obtain But, is discontinuous at As a matter of fact that. Hence, from equation(3) we have

DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS (1)Find the Fourier series of the following function, which is assume to have the period, where is given by the following figures:

DEPARTMENT OF MATHEMATICS, CVRCE Solution Here, Fourier series of the given function is where

DEPARTMENT OF MATHEMATICS, CVRCE Finally,

DEPARTMENT OF MATHEMATICS, CVRCE Hence, eqn.(1) becomes which is the required result.

DEPARTMENT OF MATHEMATICS, CVRCE (2) Find the Fourier series of the following function Solution The Fourier series is given by the formula as under where

DEPARTMENT OF MATHEMATICS, CVRCE Finally, Which is the required result.

DEPARTMENT OF MATHEMATICS, CVRCE (3) Find the Fourier series of the following function Solution The Fourier series is given by the formula where,

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and

DEPARTMENT OF MATHEMATICS, CVRCE Hence, the required Fourier series is

DEPARTMENT OF MATHEMATICS, CVRCE (4) If Show that its Fourier series Solution

DEPARTMENT OF MATHEMATICS, CVRCE where, The Fourier series is given by the formula

DEPARTMENT OF MATHEMATICS, CVRCE Finally,

DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of a’s and b’s in eqn.(1), we get which is the required result.

DEPARTMENT OF MATHEMATICS, CVRCE Assignments (1) Find the Fourier series of the following function, which is assume to have the period, where is given by the following figures: (1) Find the Fourier series of the following function, which is assume to have the period, where is given by the following figures: (i)

DEPARTMENT OF MATHEMATICS, CVRCE (ii)

DEPARTMENT OF MATHEMATICS, CVRCE (2) Find the Fourier series of the following functions, which is assumed to have the period (3) If then find the Fourier series of. Also deduced that