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Let f(x) = cos(2), x=(0,x). (a) Find the Fourier COSINE series of f(x). (b) Use the convergence to evaluate the Fourier COSINE series at x = 5m/3. (The result should be a number, not an infinite sum.) Hint: (a) Evaluate the coefficient formula of the Fourier COSINE series directly. We may need the results below to evaluate cos() and sin(). exp(i(mn+a))= exp(int)exp(ia) = (-1)" (cos(a)+isin(a)) cos(n+a)=(-1) cos(a), sin(n+a)=(-1)" sin(a) (b) f(x) = cos(x/2) is valid only in (0, π). Note that 5m/3 is outside (0, 1). We need to recall how function f(x) is extended in the Fourier COSINE series. ==>

Let f(x) = cos(2), x=(0,x). (a) Find the Fourier COSINE series of f(x). (b) Use the convergence to evaluate the Fourier COSINE series at x = 5m/3. (The result should be a number, not an infinite sum.) Hint: (a) Evaluate the coefficient formula of the Fourier COSINE series directly. We may need the results below to evaluate cos() and sin(). exp(i(mn+a))= exp(int)exp(ia) = (-1)" (cos(a)+isin(a)) cos(n+a)=(-1) cos(a), sin(n+a)=(-1)" sin(a) (b) f(x) = cos(x/2) is valid only in (0, π). Note that 5m/3 is outside (0, 1). We need to recall how function f(x) is extended in the Fourier COSINE series. ==>

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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Let f(x) = cos(2), x = (0,r). (a) Find the Fourier COSINE series of f(x). (b) Use the convergence to evaluate the Fourier COSINE series at x = 5π/3. (The result should be a number, not an infinite sum.) Hint: (a) Evaluate the coefficient formula of the Fourier COSINE series directly. We may need the results below to evaluate cos() and sin(). exp(i(mn+a))= exp(int)exp(ix)=(-1)" (cos(a)+isin(x)) ==> cos(m+a)=(-1)" cos(a), sin(n+a)=(-1)" sin(x) (b) f(x) = cos(x/2) is valid only in (0, π). Note that 5m/3 is outside (0, π). We need to recall how function f(x) is extended in the Fourier COSINE series.
Transcribed Image Text:Let f(x) = cos(2), x = (0,r). (a) Find the Fourier COSINE series of f(x). (b) Use the convergence to evaluate the Fourier COSINE series at x = 5π/3. (The result should be a number, not an infinite sum.) Hint: (a) Evaluate the coefficient formula of the Fourier COSINE series directly. We may need the results below to evaluate cos() and sin(). exp(i(mn+a))= exp(int)exp(ix)=(-1)" (cos(a)+isin(x)) ==> cos(m+a)=(-1)" cos(a), sin(n+a)=(-1)" sin(x) (b) f(x) = cos(x/2) is valid only in (0, π). Note that 5m/3 is outside (0, π). We need to recall how function f(x) is extended in the Fourier COSINE series.
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