Let f(x) = π − x, 0 ≤ x ≤ π. Sketch two periods
of
the pointwise limit of its Fourier cosine series (FCS). Is the
FCS
uniformly convergent? Why or why not?
Let f(x) = {0 if -π < x < 0 x if 0 < x < π (a) Find the Fourier series of f. (b) Sketch the graph of function to which the series converges pointwise on R. Justify your answer (c) Show that
Let f(x) = 1, 0 〈 x 〈 π. Find the Fourier cosine series with period 2T. Let f(x) = 1, 0 〈 x 〈 π. Find the Fourier sine series with period 2T.
Let f(x) = x.a) Expand f(x) in a Fourier cosine series for 0 ≤ x ≤ π.b) Expand f{x) in a Fourier sine series for 0 ≤ x < π.c) Expand fix) in a Fourier cosine series for 0 ≤ x ≤ 1.d) Expand fix) in a Fourier sine series for 0 ≤ x < 1.
4. Let f(x) = 6-2x, 0<x 2 (a) Expand f(x) into a periodic function of period 2, ie. construct the function F(x), such that F(x)-f (x), 0xS 2, and Fx) F(x+2) for all real numbers x. (This process is called the "full-range expansion" of f(x) into a Fourier series.) Find the Fourier series of Fr). Then sketch 3 periods of Fx). (b) Expand fx) into a cosine series of period 4. Find the Fourier series and sketch 3 periods (c)...
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
Problem 1. Expand f(x) em 1. Expand fo) (1.0 ,-π < x < 0 0, 0<X<T in a sine, cosine Fourier series. write out a few 0, 0<x<π in sine,cosine Fourier series Write out aferw terms of the series
Find the required Fourier Series for the given function f(x). Sketch the graph of f(x) for three periods. Write out the first five nonzero terms of the Fourier Series. cosine series, period 4 f(0) = 3 if 0<x<1, if 1<x<2 1,
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
Let f(x) be the 27-periodic function which is defined by f(x)-cos(x/4) for-π < x < 1. π. (a) Draw the graph of y f(x) over the interval-3π < x < 3π. Is f continuous on R? (b) Find the trigonometric Fourier Series (with L π) for f(x). Does the series converge absolutely or conditionally? Does it converge uniformly? Justify your answer. (c) Use your result to obtain explicit values for these three series: 16k2 1 16k2 1 (16k2 1)2 に1...
π. Compute (25) 4. Let f be the constant function f(x) = 3 defined on the interval 0くエ the Fourier sine series of f(x) on 0 x π