There are 3 questions on this assignment. The marks awarded for each part are indi- cated in boxes. 1. Consider the fun...
1. Consider the function defined by 1- x2, 0< |x| < 1, f(x) 0, and f(r) f(x+4) (a) Sketch the graph of f(x) on the interval -6, 6] (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 2. Consider the function 0 T/2, T/2, T/2 < T. f(x)= (a) Sketch the odd and even periodic extension of f(x) for -3r < x < 3m. (b) Find the Fourier cosine series...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks 3. Consider the function defined by...
1. Consider the function defined by f(x) 0, |x| < 2 1 and f(x) f(x 4) (a) Sketch the graph of f(x) on the interval -6,6 8 (b) Find the Fourier series representation of f(z). You must show how to evaluate any integrals that are needed 1. Consider the function defined by f(x) 0, |x|
1. Consider the function defined by 1-2, 0 < Ixl < 1, f(x) = and f(x) = f(x + 4). 1 (a) Sketch the graph of f(x) on the interval [-6,6). 8 (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 1. Consider the function defined by 1-2, 0
1. Consider the function defined by 1-2, 0 < Ixl < 1, f(x) = and f(x) = f(x + 4). 1 (a) Sketch the graph of f(x) on the interval [-6,6). 8 (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 1. Consider the function defined by 1-2, 0
Please solve for part (b) and (c) thank you! 1. Consider the function f(x) = e-x defined on the interval 0 < x < 1. (a) Give an odd and an even extension of this function onto the interval -1 < x < 1. Your answer can be in the form of an expression, or as a clearly labelled graph. [2 marks] (b) Obtain the Fourier sine and cosine representation for the functions found above. Hint: use integration by parts....
Consider the function 0<x<π/2. z, f(x) = (a) Sketch the odd and even periodic extension of f(x) for-3π 〈 x 〈 3π. (b) Find the Fourier cosine series of the even periodic extension of f(x) Consider the function 0
Sketch the function with its (a) odd periodic extension and (b) even then find the Fourier Sine and Fourier Cosine series, respectively. periodic extension, 0< x < X f(x) = -< x< 2 2 Sketch the function with its (a) odd periodic extension and (b) even then find the Fourier Sine and Fourier Cosine series, respectively. periodic extension, 0
3. Consider the periodic function defined by sin(x f(x)-く 0T and f(x)-f(x + 27). 1 (a) Sketch f(x) on the interval-3π 〈 3T. 9 (b) Find the complex Fourier series of f(x) and obtain from it the regular Fourier series. 3. Consider the periodic function defined by sin(x f(x)-く 0T and f(x)-f(x + 27). 1 (a) Sketch f(x) on the interval-3π 〈 3T. 9 (b) Find the complex Fourier series of f(x) and obtain from it the regular Fourier series.
4. [15 marks] Consider the function h(x) = cos(x) on x = [0,1]. (a) Sketch the even and odd periodic extensions of the function over the interval (-4,2). (b) Write both the Fourier sine and cosine series for this function. (c) Using Matlab or similar, plot the function and both Fourier series using 10 terms of the full interval on the same axes and compare. Comment on whether the convergence of both series is in line with expectation.