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(b) Derive the spectral representation by utilizing the identity. 1 2πi & G(x, 5; 2) da CR 8(x - 5): where == (c) Obtain the Fourier transform pair from the result of (b): 8 f(x) = f f(x) elkx dk 88 8 F(x) = 2 / f(x) e-kx dx. f(k): 2π 88 We call f(k) a Fourier transform of f(x).

(b) Derive the spectral representation by utilizing the identity. 1 2πi & G(x, 5; 2) da CR 8(x - 5): where == (c) Obtain the Fourier transform pair from the result of (b): 8 f(x) = f f(x) elkx dk 88 8 F(x) = 2 / f(x) e-kx dx. f(k): 2π 88 We call f(k) a Fourier transform of f(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Consider an eigenvalue problem -u" - Au = 0; uES = C₂ (-00,00) and VvE S, v(-00)&v(0) = ok. (a) Determine the Green's function G(x, ; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 27/1₁ & G(x, 5; 2) da if 2πί CR (c) Obtain the Fourier transform pair from the result of (b): where 8(x - 5) = f(x) = [ f(x) e¹kx dk ƒ (k) = 1 2π [ f(x) e-ikx dx. 88 We call f(k) a Fourier transform of f(x).
Transcribed Image Text:5. Consider an eigenvalue problem -u" - Au = 0; uES = C₂ (-00,00) and VvE S, v(-00)&v(0) = ok. (a) Determine the Green's function G(x, ; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 27/1₁ & G(x, 5; 2) da if 2πί CR (c) Obtain the Fourier transform pair from the result of (b): where 8(x - 5) = f(x) = [ f(x) e¹kx dk ƒ (k) = 1 2π [ f(x) e-ikx dx. 88 We call f(k) a Fourier transform of f(x).
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