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- Consider the Fourier expansion = (−1)²k+1 . 2 k ∞ f(x) = [ck · sin(kx), k=1 where Ck = Describe the periodic function f(x). Verify that the first Fourier coefficient satisfies C1 = CTT -π xsin (x) dx.-3/2 J512 (x)dx = (b) Prove that 2 COS X sin x x-2 Jy2 (x)dx = + C. %3D x2 x31. 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.
- 1. Calculate Fourier coefficients: Let f(t) = sin 2nt - cos 2nt +5 sin 6xt. (a) Use the integral formula to compute f(n) for all n. (b) Compute the Fourier expansion of f without using integrals (i.e. via identities) to verify your answer.Please read the question and leave notes on the answer where appropriate. Please DO NOT skip any steps. Please double check your answer. DO NOT SUBMIT A TYPED RESPONSE. Please use a hand written response that is LEGABLE. Please review your answer to make sure all steps are clearly visible. Please double check your work. Thank you. Show all your work and please make sure to Justify your answers. I am having trouble finding this answer. Please double check your steps and do not submit a typed response that is not in LATEX.what is the Fourier expansion of the periodic function. F(x)= -2Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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