tailieunhanh - Ebook Theory and problems of fourier analysis with applications to boundary value problems: Part 2

(BQ) Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. | chapter 5 Fourier Integrals and Applications THE NEED FOR FOURIER INTEGRALS In Chapter 2 we considered the theory and applications involving the expansion of a function f x of period 2L into a Fourier series. One question which arises quite naturally is what happens in the case where L - 00 We shall find that in such case the Fourier series becomes a Fourier integral. We shall discuss Fourier integrals and their applications in this chapter. THE FOURIER INTEGRAL Let us assume the following conditions on f x 1. f x and f x are piecewise continuous in every finite interval. 2. I dx converges . f x is absolutely integrable in 00 oo . z 00 Then Fourier s integral theorem states that A a cos aX B a sin ax da 0 1 r A a 3 I f x cos aX dx 7T 00 where If . B a - I f x sin ax dx 7T -00 Ơ 2 The result 1 holds if a is a point of continuity of fix . If a is a point of discontinuity .________. f x 0 f x - 0 ________. ______. x . we must replace f x by -------- -------- as in the case of Fourier series. Note that the above conditions are sufficient but not necessary. The similarity of 1 and 2 with corresponding results for Fourier series is apparent. The right-hand side of 1 is sometimes called a Fourier integral expansion of f x . EQUIVALENT FORMS OF FOURIER S INTEGRAL THEOREM Fourier s integral theorem can also be written in the forms f eiax da du ẩ 4 0 eia z- du da 80 CHAP. 5 FOURIER INTEGRALS AND APPLICATIONS 8Í where it is understood that if f x is not continuous at X the left side must be replaced by f x 0 f x-0 These results can be simplified somewhat if f x is either an odd or an even function and we have 2 f - 1 sin aX da z 00 1 f u sin au du 0 if f x is odd 5 f x 2 p 1 cos aX da 7T 7o s 00 1 fịụ cos au du 0 if f x IS even ff FOURIER TRANSFORMS From 4 it follows that if F e u du 1 r then f x I F 0 eiax da úilĩ J 00 7 S The function F a is called the Fourier transform of f x and is sometimes written F a F f x . The function f x is the inverse Fourier transform of F a .

• Toán học
• Fourier analysis
• Boundary value problems
• Theory fourier analysis
• Problems fourier analysis
• Fouried integrals
• Bessel functions
• Orthogonal functions
• Lecture Physiological signal processing
• Bài giảng Xử lý tín hiệu sinh lý
• Physiological signal processing
• Discrete fourier transform
• Computational fourier analysis
• Lecture ELEC2400 signals and systems
• ELEC2400 signals and systems
• Discrete time fourier analysis
• Discrete time fourier transform
• Advanced Algorithms Analysis and Design
• Lecture Advanced Algorithms Analysis and Design
• Bài giảng Phân tích và thiết kế thuật toán nâng cao
• Fast fourier transform
• Coefficient representation
• Bài giảng Xử lý tín hiệu số
• Xử lý tín hiệu số
• Fourier Transform
• Frequency analysis of discrete time signal
• Properties of Fourier transform
• Frequency domain characteristics of LTI systems
• Time series
• Computer algorithm
• Harmonic model
• Lomb periodogram
• Fourier Series
• Orbitrap Instruments
• Cotton Trash
• Molecular Simulation
• Surface Chemistry
• Design and Analysis of Algorithms
• Analysis of Algorithms
• Design of Algorithms
• Data strutures
• Polynomial operations
• Conquer algorithm
• Applications in Nanomaterials
• FTICR and Orbitrap
• Cotton and Cotton Trash
• Exponential Type Functions
• DISCRETE TIME SIGNALS
• THE DISCRETE TIME FOURIER ANALYSIS
• THE Z TRANSFORM
• THE DISCRETE FOURIER TRANSFORM
• DIGITAL FILTER STRUCTURES
• FIR FILTER DESIGN
• Electric circuits
• AC power
• The laplace transform
• Three phase systems
• Circuit analysis
• Magnetically coupled circuits
• Digital Signals
• Sampling
• Basic Types
• Time domain Analysis
• Frequency Domain Analysis
• Z Domain Analysis
• Spectral Analysis
• Critical speed and natural frequency analysis of cracked rotor shaft
• Cracked rotor shaft
• Critical speed of cracked rotor shaft
• Natural frequency analysis of cracked rotor shaft
• BMC Bioinformatics
• Power spectrum
• Image analysis
• Artificial images
• Electronic Engineering
• Harmonic analysis
• Fourier transforms
• Power system parameter estimation
• Journal of Science and Technology
• Performance of Tio2 in photodegradation seafood wastewater
• Energy Dispersive X Ray analysis
• X ray powder diffraction analysis
• Scanning electron microscopy
• Fourier transform infrared
• Complex analysis
• Mathematics and engineering
• Residue theory
• Conformal mapping
• Harmonic functions
• Laplace transform
• Fundamental numerical methods
• Sata analys
• Numerical solution
• Integral equations
• Least squares
• Statistical tests
• Sampling distributions of moments
• BMC Genomics
• Taxonomic classification
• Whole genome analysis
• Genomic signature
• Alignment free sequence analysis
• Machine learning
• Numerical representation of DNA sequences
• Digital signal processing
• Frequency response of LTI systems
• Journal of Zoology
• Fourier transform infrared spectroscopy
• Thermogravimetric analysis
• Peripatoides novaezealandiae group
• Single analysis X ray diffraction
• MATLAB BASIC
• PLOTTING COMMANDS
• CONTROL STATEMENTS
• NETWORK FUNCTIONS
• DIODES
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