Consider a periodic function f(x) defines as follows: 4. f(x)-0 f(x)-0 The function is periodic e...
Consider a periodic function f(x) defines as follows: -π < x < -π/2, f(x) = 0 -π/2 < x < π/2, f(x) = 1 π/2 < x < π, f(x) = 0 The function is periodic every 2π. Find the first four non-zero terms in the Fourier series of this function for the interval [-π, π] or equivalently for the interval [0, 2π]. Note that depending if the function is odd or even, the first four terms do not necessarily...
5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0 < x < π. (Only sketch over the interval z E [-2π, 2π). (b) (10) Find the Fourier sine series of the function in part (a) 5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0
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
Consider the function f (x) = cos x, 0 < x < π. (a) Find the Fourier series of the periodic odd extension of f. (b) State the interval in which the series in (a) converges to f.
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...
Fourier Series for Odd Functions Recall that if f is an odd function, f(-x)f(x). An odd Fourier series has only the sine terms, and can be approximate an odd function, so Fo(x) b sinx)+b2 sin(2x)+ b, sin(3x)+. Why is there no b, term in the series F, (x)? 1. 2. Using steps similar to those outlined for even functions, develop a rule for finding the coefficients to approximate any odd function on the interval [-π, π]. 3. If f (x)sin...
4. Consider the periodic function given below: f(x)-x 0 ㄨㄑㄧ (i) State its fundamental period, and sketch the function for 3 periods. (5 marks) i) Find the Fourier series of the given periodic function, and expand the series to give the first three non-zero a and b terms (15 marks) ii) Use the answer obtained in Q4(ii) and the given periodic function, find the sum of the series 4(2n-1 )2 (5 marks)
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
Consider a periodic function f(x) given as -7, f(x) = { - < x < 0, 0 < x <, TT – I, f(x) = f(x + 27). i) Sketch the graph of f(x) in the interval –37 < x < 37. Then, deter- mine whether f(x) is even, odd or neither. (3 marks) ii) Hence, find the Fourier series of f(x). (12 marks)
For the function y 1-x for 0 s x s 1 Graph the function's 3 periods 1) Find its formulas for the Fourier series and Fourier coefficients 2) Write out the first three non-zero terms of the Fourier Series 3) 4) Graph the even extension of the function 5) Find the Fourier series and Fourier coefficients for the even extension 6) Write out the first three non-zero terms of the even Fourier series 7) Graph the odd extension of the...