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

MATHEMATICS-II LECTURE-15 Half Range Fourier Series Expansions [Chapter – 10.4] Half Range Fourier Series Expansions [Chapter – 10.4] DEPARTMENT OF MATHEMATICS, CVRCE TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION]

Definition Fourier series as half range expansions Let us consider the Fourier series expansion of a function which is periodic, defined in an interval of length 2L. Now we wish to expand a non-periodic function defined in half of the above interval say 0 < x < L of length L, such an expansions are known as half range expansions or half range Fourier series. Let us consider the Fourier series expansion of a function which is periodic, defined in an interval of length 2L. Now we wish to expand a non-periodic function defined in half of the above interval say 0 < x < L of length L, such an expansions are known as half range expansions or half range Fourier series. DEPARTMENT OF MATHEMATICS, CVRCE

 A half range expansion containing only cosine term is known as half range Fourier cosine series of in the interval.  A half range expansion containing only cosine term is known as half range Fourier cosine series of in the interval.  A half range expansion containing only sine term is known as half range Fourier sine series of in the interval.  A half range expansion containing only sine term is known as half range Fourier sine series of in the interval.

Fourier cosine series as the half range When we wish to express the function as a cosine series in the interval, then we extend the function reflecting it in the y -axis, so that which is symmetrical about y - axis. Then the extended function is even in the interval and its expansion will give the required Fourier cosine series which is given by the formula as: DEPARTMENT OF MATHEMATICS, CVRCE

with the Fourier coefficients DEPARTMENT OF MATHEMATICS, CVRCE Fourier cosine series as the half range

Fourier sine series as the half range To express the function as a sine series in the interval, then we extend the function reflecting it in the origin, so that which is symmetrical about the origin. Then the extended function is odd in the interval And the expansion will give the required Fourier sine series which is given by the formula as: To express the function as a sine series in the interval, then we extend the function reflecting it in the origin, so that which is symmetrical about the origin. Then the extended function is odd in the interval And the expansion will give the required Fourier sine series which is given by the formula as: DEPARTMENT OF MATHEMATICS, CVRCE

Fourier sine series as the half range with the Fourier coefficient

Some Solved Problem Based on Half Range Expansion DEPARTMENT OF MATHEMATICS, CVRCE 1. Find the Fourier cosine series as well as the Fourier sine series of the following function: Solution : The graph of

We know that the half range Fourier cosine series of the given function f(x) in the prescribed range is given by with the Fourier coefficients DEPARTMENT OF MATHEMATICS, CVRCE Some Solved Problem Based on Half Range Expansion

DEPARTMENT OF MATHEMATICS, CVRCE Some Solved Problem Based on Half Range Expansion

Substituting the values of a n ’s in the eqn.(1), we get which is the required Fourier cosine series of f(x) over the half range 0<x<L. DEPARTMENT OF MATHEMATICS, CVRCE Some Solved Problem Based on Half Range Expansion

DEPARTMENT OF MATHEMATICS, CVRCE The graph of f(x)=x in 0<x<L is the line OA. Let us extend the function f(x) in the interval –L<x<0, shown by the line OA’, so that the new function is symmetrical about the origin which represents an odd function in –L<x<L that contain sine series only and is given by the formula as: with the Fourier coefficient Some Solved Problem Based on Half Range Expansion

DEPARTMENT OF MATHEMATICS, CVRCE

Putting the values of b n ’s in eqn.(2), we obtain the desired Fourier sine series over the half range 0 < x < L as Some Solved Problem Based on Half Range Expansion DEPARTMENT OF MATHEMATICS, CVRCE

Some Solved Problem Based on Half Range Expansion 2. Find the Fourier cosine series as well as the Fourier sine series of the following function. DEPARTMENT OF MATHEMATICS, CVRCE Solution : We know that the Fourier cosine series as a half range is given by We know that the Fourier cosine series as a half range is given by

DEPARTMENT OF MATHEMATICS, CVRCE with the Fourier coefficients Some Solved Problem Based on Half Range Expansion and

DEPARTMENT OF MATHEMATICS, CVRCE Some Solved Problem Based on Half Range Expansion Substituting the values of a n ’s in eqn.(1), we get is the required half range Fourier cosine series of f(x).

DEPARTMENT OF MATHEMATICS, CVRCE We know that the Fourier sine series is given by Some Solved Problem Based on Half Range Expansion where

Substituting the values of b n ’s in eqn.(1), we get is the required half range Fourier sine series of f(x). DEPARTMENT OF MATHEMATICS, CVRCE Some Solved Problem Based on Half Range Expansion

3. Find the half-range Fourier sine series of Solution: where We know that the Fourier sine series is given by

Some Solved Problem Based on Half Range Expansion

Substituting the values of b n ’s in eqn.(1), we get is the required half range Fourier sine series of f(x). Some Solved Problem Based on Half Range Expansion

Assignments (1) Find the Fourier cosine series as well as the Fourier sine series of the following functions: (2) Find the two half-range expansions of the functions DEPARTMENT OF MATHEMATICS, CVRCE

Assignments Contd … (3) Find the half-range Fourier sine and cosine series for the function as DEPARTMENT OF MATHEMATICS, CVRCE