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(a) Consider the following function and its periodic extension: f(x) = sin z], where and F(x) = f(x+nx), Find the Fourier series representation of F(x). Note that L- म where REZ

(a) Consider the following function and its periodic extension: f(x) = sin z], where and F(x) = f(x+nx), Find the Fourier series representation of F(x). Note that L- म where REZ

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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(a) Consider the following function and its periodic extension: π f(x) = sinx, where M <1 Find the Fourier series representation of F(x). Note that L = and F(x) = f(x+nn). where NEZ 0x0 Σ n=1 (b) Does the Fourier series you found in (a) converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (c) Use your result in (a) to evaluate the following two series 1 4n² 1 and π 2 n=1 (-1)" 4n² 1 (d) Attempt to differentiate your Fourier series in (a) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (e) Carefully differentiate the function F(x), as defined in (a), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F'(z). (f) Write 1-2 sentences comparing your series' from (d) and (e), discussing how well the term by term differentiation has worked. (g) Attempt to differentiate your Fourier series in (d) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (h) Carefully differentiate your answer from (e), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F"(x). You may reuse working from previous questions. (i) Write 1-2 sentences comparing your series from (g) and (h), discussing how well the term by term differentiation has worked.
Transcribed Image Text:(a) Consider the following function and its periodic extension: π f(x) = sinx, where M <1 Find the Fourier series representation of F(x). Note that L = and F(x) = f(x+nn). where NEZ 0x0 Σ n=1 (b) Does the Fourier series you found in (a) converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (c) Use your result in (a) to evaluate the following two series 1 4n² 1 and π 2 n=1 (-1)" 4n² 1 (d) Attempt to differentiate your Fourier series in (a) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (e) Carefully differentiate the function F(x), as defined in (a), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F'(z). (f) Write 1-2 sentences comparing your series' from (d) and (e), discussing how well the term by term differentiation has worked. (g) Attempt to differentiate your Fourier series in (d) term by term. Does the resulting series converge uniformly? If so, prove this using the Weierstrass test for uniform convergence. If not, discuss whether it converges pointwise anywhere in its domain. (h) Carefully differentiate your answer from (e), and then find the Fourier series of the resulting function, i.e., find the Fourier series of F"(x). You may reuse working from previous questions. (i) Write 1-2 sentences comparing your series from (g) and (h), discussing how well the term by term differentiation has worked.
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