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The Fourier expansion for any (reasonable) function m f(x) = 40 + an (ΐ π) + Σón sin n=1 n=1 Lancos ( (a) Given this expansion, prove that aj = 1 L r+L Y +L C 1 b₁ = dx f(x) sin(a L dx f(x) cos -L (2) L

The Fourier expansion for any (reasonable) function m f(x) = 40 + an (ΐ π) + Σón sin n=1 n=1 Lancos ( (a) Given this expansion, prove that aj = 1 L r+L Y +L C 1 b₁ = dx f(x) sin(a L dx f(x) cos -L (2) L

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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1. The Fourier expansion for any (reasonable) function may be written f(x) = 2 + Žª, cos (¹72) + [b, sin (7²). Σ n=1 (a) Given this expansion, prove that +L = 1/² = L ·L r+L b;="dzf(x) sin (47) (1₁). L aj jπ dx f(x) cos X L (While it doesn't affect the math, use L = π for convenience.) (b) Show that for even functions (f(-x) = f(x)), bn = 0 for all n, and that for odd functions (f(-x) = -f(x)), an = 0 for all n.
Transcribed Image Text:1. The Fourier expansion for any (reasonable) function may be written f(x) = 2 + Žª, cos (¹72) + [b, sin (7²). Σ n=1 (a) Given this expansion, prove that +L = 1/² = L ·L r+L b;="dzf(x) sin (47) (1₁). L aj jπ dx f(x) cos X L (While it doesn't affect the math, use L = π for convenience.) (b) Show that for even functions (f(-x) = f(x)), bn = 0 for all n, and that for odd functions (f(-x) = -f(x)), an = 0 for all n.
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