Find the Fourier coefficients of f(z)=7z-6. That is, find the numbers a, a, and b (where a, and by may depend on k) such thatf(z)= ao+a1 cos(a) + by sin(z) + a₂ cos(2x) + b₂ sin(22) +...for all z with-T

Given f(x)=7x-6 over x∈(-π,π) Fourier coefficients: a0=12π∫-ππf(x)dx=12π∫-ππ[7x-6]dx…

Answered: Find the Fourier coefficients of…

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.
Find the cosine Fourier coefficients for the function f(x) = 35 cos(3x), on 0Sxs. O A None of these. 140 and an- 35 (-1)+1 (-1)" 2n+3 2n-3 O B.40 =- 37 n=1,2,3,... 140 and an- 70 (-1)+1 (-1)" 2n +3 Oca0 =- 2n-3 n=1,2,3,... 140 and an= 元 70 (-1)*1 2n +3 (-1)" 2n-3 O Da0 =- n = 1,2,3,... %3D 70 (-1)+1 (-1)" 2n-3 2n+3 n= 0,1,2,3,... OE an=
2. Determine the complex Fourier coefficients co and c„ of the function simulating the periodic heating of a room if it is given that f(t) = e' – 1, 0
Suppose that f(t) is periodic with period [-T, 7) and has the following complex Fourier coefficients: -4, C1 —3+ 41, С2 —D —1- 21, C3 2. (A) Compute the following complex Fourier coefficients. C_3 = 2 C_2 =-1+2i C_1 = (B) Compute the real Fourier coefficients. (Remember that ei kt cos(kt) + i sin(kt).) ao = a1 = -6 , a2 = -2 , az = 4 b, = b2 = b3 = (C) Compute the complex Fourier coefficients of the following. (i) The derivative f' (t). Co = C1 = -4-31 C2 = C3 = (ii) The shifted function f(t + Co = C = C2 = C3 = (iii) The function f(3t). Co = C = C2 = C3 =
(b) Determine the Z-transforms of the following expressions (ii) (cos 0 + i sin 0)"
b) Find the order of Z=0 for the 25 1) 1+2-e² following functions: 3) (e ²²-1-2²) Sin³z 2) 6 sin(z ³) + z⁹-6z³
find the fourier trasnforms of the the function provided. Also can you write an explanantion of th eprocess to understand better.
44 Ex. 1. Prove that : 2F ( ½ ‚ ½ ; ³ ; x² ) = ÷ sin¯¹ x } + (a) 24 (6) 2³1 ( = ¹ i ² i-x²) = -— (an²¹1 x ,1 -tan-1 2 X of hypergeometric functi
9
6. Let 2, = 2| cos + i sin 3 3, 22 = + i sin 4 and cos 4 (a) Find the product 2,22- (b) Find the quotient z1/z3.
b) Find the inverse z-transform of the following: i. z-15 z|2| sin 3 z²-(2|2|cos 3)z+|2|2. ii. z-12 (z+4z+1) 1 (z-1)4 4 iii. z-1 z(z+1*) 2 (z-13)3
Please answer part (a) only

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Find the Fourier coefficients of f(z)=7z-6. That is, find the numbers a, a, and by (where a, and by may depend on k) such that f(z)= ao+acos(z) + by sin(z) + a, cos(22) + b₂ sin(2z) +... for all z with-CNCH. ag= afm by = if k is odd, and if k is odd, and it k is even, if k is even.