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

MATHEMATICS - II ● LAPLACE TRANSFORMS ● FOURIER SERIES FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING] ● LAPLACE TRANSFORMS ● FOURIER SERIES ● FOURIER TRANSFORMS ● VECTOR DIFFERENTIAL CALCULUS ● VECTOR INTEGRAL CALCULUS ● LINE, DOUBLE, SURFACE, VOLUME INTEGRALS ● BETA AND GAMMA FUNCTIONS TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8th EDITION] DEPARTMENT OF MATHEMATICS, CVRCE

FOURIER INTEGRAL [chapter – 10.8] MATHEMATICS - II LECTURE :16 DEPARTMENT OF MATHEMATICS, CVRCE

LAYOUT OF LECTURE FROM FOURIER SERIES TO FOURIER INTEGRAL INTRODUCTION & MOTIVATION APPLICATIONS EXISTENCE OF FOURIER INTEGRAL SOME PROBLEMS FOURIER SINE AND COSINE INTEGRALS DEPARTMENT OF MATHEMATICS, CVRCE

INTRODUCTION & MOTIVATION FOURIER SERIES ARE POWERFUL TOOLS IN TREATING VARIOUS PROBLLEMS INVOLVING PERIODIC FUNCTIONS. HOWEVER, FOURIER SERIES ARE NOT APPLICABLE TO MANY PRACTICAL PROBLEMS SUCH AS A SINGLE PULSE OF AN ELECTRICAL SIGNAL OR MECHANICAL FORCE VIBRATION WHICH INVOLVE NONPERIODIC FUNCTIONS. THIS SHOWS THAT METHOD OF FOURIER SERIES NEEDS TO BE EXTENDED. HERE WE START WITH THE FOURIER SERIES OF AN ARBITRARY PERIODIC FUNCTION fL OF PERIOD 2L AND THEN LET L SO AS TO DEVELOP FOURIER INTEGRAL OF A NON-PERIODIC FUNCTION. Jean Baptiste Joseph Fourier (Mar21st 1768 –May16th 1830) French Mathematician & Physicist

FROM FOURIER SERIES TO FOURIER INTEGRAL Let fL (x) be an arbitrary periodic function whose period is 2L which can be represented by its Fourier series as follows: DEPARTMENT OF MATHEMATICS, CVRCE

FROM FOURIER SERIES TO FOURIER INTEGRAL From (1), (2), (3), and (4), we get

FROM FOURIER SERIES TO FOURIER INTEGRAL The representation (6) is valid for any fixed L, arbitrary large but finite. We now let L and assume that the resultant non-periodic function is absolutely integrable on x-axis , i.e. the resulting non-periodic function

FROM FOURIER SERIES TO FOURIER INTEGRAL is absolutely integrable. Set

FROM FOURIER SERIES TO FOURIER INTEGRAL Applying (8) and (9) in (7), we get Representation (10) with A(w) and B(w) given by (8) and (9), respectively, is called a Fourier integral of f(x).

FOURIER INTEGRAL

EXISTENCE OF FOURIER INTEGRAL Theorem If a function f(x) is piecewise continuous in every finite interval and has a right-hand derivative and left-hand derivative at every point and if the integral exists, then f(x) can be represented by a Fourier integral. At a point where f(x) is discontinuous the value of the Fourier integral equals the average of the left and right hand limits of f(x) at that point.

PROBLEMS ON FOURIER INTEGRAL EXAMPLE-1: Find the Fourier integral representation of the function Solution: The Fourier integral of the given function f(x) is

PROBLEMS ON FOURIER INTEGRAL

PROBLEMS ON FOURIER INTEGRAL

PROBLEMS ON FOURIER INTEGRAL

DIRICHLET’S DISCONTINUOUS FACTOR From Example – 1 by Fourier integral representation, we find that At x= 1, the function f(x) is discontinuous. Hence the value of the Fourier integral at 1 is ½ [f(1-0)+f(1+0)] = ½[0+1]=1/2.

DIRICHLET’S DISCONTINUOUS FACTOR The above integral is called Dirichlet’s discontinuous factor.

Sine integral Dirichlet’s discontinuous factor is given by. Putting x= 0 in the above expression we get . The integral (1) is the limit of the integral as u   The integral is called the sine integral and it is denoted by Si(u)

FOURIER COSINE INTEGRAL

FOURIER COSINE INTEGRAL The Fourier integral of an even function is also known as Fourier cosine integral.

FOURIER SINE INTEGRAL

FOURIER SINE INTEGRAL The Fourier integral of an odd function is also known as Fourier sine integral.

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL Solution: The Fourier cosine integral of the given function f(x) is

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL The Fourier sine integral of the given function f(x) is

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL Hence the Fourier sine integral of the given function is

LAPLACE INTEGRALS From Example – 2 by Fourier cosine integral representation, we find that (A)

The integrals (A) and (B) are called as Laplace integrals. From Example – 2 by Fourier sine integral representation, we find that (B) The integrals (A) and (B) are called as Laplace integrals.

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The Fourier cosine integral of the function f(x) is given by SOME MORE PROBLEMS Solution: The Fourier cosine integral of the function f(x) is given by

SOME MORE PROBLEMS

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The Fourier cosine integral of f(x) is given by SOME MORE PROBLEMS Solution: The Fourier cosine integral of f(x) is given by

SOME MORE PROBLEMS Hence from (1) we obtain the required Fourier cosine integral as

The Fourier cosine integral of the given function f(x) is SOME MORE PROBLEMS The Fourier cosine integral of the given function f(x) is

SOME MORE PROBLEMS

Hence the Fourier cosine integral of the given function is SOME MORE PROBLEMS Hence the Fourier cosine integral of the given function is

The Fourier sine integral of the given function f(x) is SOME MORE PROBLEMS The Fourier sine integral of the given function f(x) is

Hence the Fourier sine integral of the given function is SOME MORE PROBLEMS Hence the Fourier sine integral of the given function is