Half Range Fourier Series

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  Mathematical Physics

  An Introduction

  • Derek Raine(Author)
  • 2018(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  ω would have a much less discernible effect.

  Fourier methods are also commonly used in mathematical physics. In this chapter, we will focus on using them to solve differential equations, and the wave equation in particular. We will examine Fourier half range series and Fourier full range series, study some applications of Fourier series, then finish by introducing Fourier transforms and the convolution theorem.

  12.2.FOURIER HALF RANGE SINE SERIES

  In Chapter 11 , we calculated the separable solutions for a wave on a string that is fixed at both ends, at x = 0 and at x = L. In general, the displacement of such a string is

  where each of the a n and b n for n = 1, 2, 3 . . . is an arbitrary constant that we can set once we know the boundary conditions for any given problem. The range 0 to L is called the half range because it is half the maximum wavelength or spatial period.

  Consider the case when the string is initially at rest and has initial displacement y (x , 0) = f (x ). Then, by substituting t = 0 into equation (12.2), we find

  Given any physically reasonable function, f (x ), can we find the coefficients, b n , such that equation (12.3) is satisfied? Remarkably, yes! This is known as Fourier’s theorem.

  Equation (12.3) is the Fourier half range sine series of a function, f (x ). This is a very powerful result. It tells us that, within the range 0 to L , we can write any

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  Advanced Engineering Mathematics and Analysis

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Fourier Series and Fourier Transform Fourier series The first four Fourier series approximations for a square wave In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier ________________________ WORLD TECHNOLOGIES ________________________ analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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  Advanced Mathematical Analysis and Engineering Mathematics

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Fourier Series and Fourier Transform Fourier series The first four Fourier series approximations for a square wave In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier ________________________ WORLD TECHNOLOGIES ________________________ analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigen-solutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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  Advanced Engineering Mathematics, SI Edition

  • Peter O'Neil(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  AkeSak / Shutterstock.com Fourier series are the foundation for most modern imaging, signal processing, and analytical devices and algorithms. As such, they are a basic tool for the electrical engineers who design MRI and ultrasound machines for medical applications and NMR and IR spectrometers for chemical analysis. To translate the signal output from these machines into useful informa-tion for doctors and chemists, computer programmers write programs, based on the Fast Fourier Transform (FFT) algorithm, which combines the Fourier methods of this chapter with the matrix methods of previous chapters. Even broadcast and audio engineers use frequency filters based on Fourier analysis to amplify voices or cut out noise from their transmissions. Section 17.1 Problems In each of Problems 1–12, write the Fourier series for the function on the interval and determine the sum of the series. If software is available, graph some partial sums of the Fourier series. 1. f ( x ) = 4, − 3 ≤ x ≤ 3 2. f ( x ) = − x , − 1 ≤ x ≤ 1 3. f ( x ) = cosh (π x ) , − 1 ≤ x ≤ 1 4. f ( x ) = 1 − | x | , − 2 ≤ x ≤ 2 5. f ( x ) = − 4 for − π ≤ x ≤ 0, 4 for 0 < x ≤ π 6. f ( x ) = sin ( 2 x ) , − π ≤ x ≤ π 7. f ( x ) = x 2 − x + 3, − 2 ≤ x ≤ 2 8. f ( x ) = − x for − 5 ≤ x < 0, 1 + x 2 for 0 ≤ x ≤ 5 9. f ( x ) = 1 for − π ≤ x < 0, 2 for 0 ≤ x ≤ π 10. f ( x ) = cos ( x / 2 ) − sin ( x ) , − π ≤ x ≤ π 11. f ( x ) = cos ( x ) , − 3 ≤ x ≤ 3 12. f ( x ) = 1 − x for − 1 ≤ x ≤ 0, 0 for 0 < x ≤ 1. In each of Problems 13–19, determine the sum of the Fourier series on the interval. It is not necessary (or even helpful) to write this series. 13. f ( x ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2 x for − 3 ≤ x < − 2, 0 for − 2 ≤ x < 1 x 2 for 1 ≤ x ≤ 3 14. f ( x ) = 2 x − 2 for − π ≤ x < 1, 3 for 1 < x ≤ π Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 17.2 Sine and Cosine Series 597 15. f ( x ) = x 2 for − π ≤ x ≤ 0, 2 for 0 < x ≤ π 16.

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  Bird's Higher Engineering Mathematics

  • John Bird(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  (c), for a half-range Fourier sine series: f (x) = ∞ ∑ n=1 b n sin nx dx b n = 2 π ∫ π 0 f (x) sin nx dx = 2 π ∫ π 0 cos x sin nx dx = 2 π ∫ π 0 1 2 [sin(x + nx) - sin(x - nx)] dx from 7 of Table 39.1, page 468 = 1 π [ -cos[x(1 + n)] (1 + n) + cos[x(1 - n)] (1 - n) ] π 0 = 1 π [( -cos[π(1 + n)] (1 + n) + cos[π(1 - n)] (1 - n) ) - ( -cos 0 (1 + n) + cos 0 (1 - n) )] 682 Section K When n is odd, b n = 1 π [( -1 (1 + n) + 1 (1 - n) ) - ( -1 (1 + n) + 1 (1 - n) )] = 0 When n is even, b n = 1 π [( 1 (1 + n) - 1 (1 - n) ) - ( -1 (1 + n) + 1 (1 - n) )] = 1 π ( 2 (1 + n) - 2 (1 - n) ) = 1 π ( 2(1 - n) - 2(1 + n) 1 - n 2 ) = 1 π ( -4n 1 - n 2 ) = 4n π(n 2 - 1) Hence b 2 = 8 3π , b 4 = 16 15π , b 6 = 24 35π and so on. Hence the half-range Fourier sine series for f (x) in the range 0 to π is given by: f (x) = 8 3π sin 2x + 16 15π sin 4x + 24 35π sin 6x + ··· or f (x) = 8 π ( 1 3 sin 2x + 2 (3)(5) sin 4x + 3 (5)(7) sin 6x + ··· ) Now try the following Practice Exercise Practice Exercise 268 Half-range Fourier series (Answers on page 902) 1. Determine the half-range sine series for the function defined by: f (x) =    x, 0 < x < π 2 0, π 2 < x < π 2. Obtain (a) the half-range cosine series and (b) the half-range sine series for the function f (t) =      0, 0 < t < π 2 1, π 2 < t < π 3. Find the half-range Fourier sine series for the function f (x) = sin 2 x in the range 0 ≤ x ≤ π. Sketch the function within and outside of the given range. 4. Determine the half-range Fourier cosine series in the range x = 0 to x = π for the function defined by: f (x) =        x, 0 < x < π 2 (π - x), π 2 < x < π Practice Exercise 269 Multiple-choice questions on even and odd functions and half-range Fourier series (Answers on page 902) Each question has only one correct answer 1.

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  Mathematical Methods for Physicists

  A Concise Introduction

  • Tai L. Chow(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  The Dirichlet conditions are generally satisfied in practice. Half-range Fourier series Unnecessary work in determining Fourier coecients of a function can be avoided if the function is odd or even. A function f x † is called odd if f x † ˆ f x † and even if f x † f x † ˆ f x † . It is easy to show that in the Fourier series corresponding to an odd function f o x † , only sine terms can be present in the series expansion in the interval < x < , for a n ˆ 1 Z f o x † cos nx dx ˆ 1 Z 0 f o x † cos nx dx ‡ Z 0 f o x † cos nx dx ˆ 1 Z 0 f o x † cos nx dx ‡ Z 0 f o x † cos nx dx ˆ 0 n ˆ 0 ; 1 ; 2 ; . . . ; 4 : 6a † 151 HALF-RANGE FOURIER SERIES Figure 4.6. A piecewise continuous function. but b n ˆ 1 Z 0 f o x † sin nx dx ‡ Z 0 f o x † sin nx dx ˆ 2 Z 0 f o x † sin nx dx n ˆ 1 ; 2 ; 3 ; . . . : 4 : 6b † Here we have made use of the fact that cos( nx † ˆ cos nx and sin nx † ˆ sin nx . Accordingly, the Fourier series becomes f o x † ˆ b 1 sin x ‡ b 2 sin 2 x ‡ : Similarly, in the Fourier series corresponding to an even function f e x † , only cosine terms (and possibly a constant) can be present. Because in this case, f e x † sin nx is an odd function and accordingly b n ˆ 0 and the a n are given by a n ˆ 2 Z 0 f e x † cos nx dx n ˆ 0 ; 1 ; 2 ; . . . : 4 : 7 † Note that the Fourier coecients a n and b n , Eqs. (4.6) and (4.7) are computed in the interval (0, ) which is half of the interval ( ; ). Thus, the Fourier sine or cosine series in this case is often called a half-range Fourier series. Any arbitrary function (neither even nor odd) can be expressed as a combina-tion of f e x † and f o x † as f x † ˆ 1 2 f x † ‡ f x † ‰ Š ‡ 1 2 f x † f x † ‰ Š ˆ f e x † ‡ f o x † : When a half-range series corresponding to a given function is desired, the function is generally defined in the interval (0, ) and then the function is specified as odd or even, so that it is clearly defined in the other half of the interval ; 0 † .

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