Homework Answers
The reason for this to be False is because the function f will
not converge to its value at a particular point c if it is not
continuous at point c. The correct statement is that Fourier series
will converge to the value of the function f(c) at all those points
where f is continuous and will converge value ![[lim-()lim>ctf()]](http://img.homeworklib.com/images/c9635e10-a102-4fdc-87df-2ac445a0ee93.latex?%5Clarge%20%5Cfrac%7B1%7D%7B2%7D%5C%3A%20%5Blim_%7Bx-%3Ec%5E-%7Df%28x%29+lim_%7Bx-%3Ec%5E+%7Df%28x%29%5D?x-oss-process=image/resize,w_560)
where the terms in the bracket are the left and right limit of
the function f at the point of discontinuity 
Thus continuity of the function f is vital in convergence of the function and the value attained by the function.
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Why is this one false? False A Fourier series will converge to the value of the function at all points if the function...
b) Consider the function g : [-π, π] → R, g(z) = -1 otherwise Does the Fourier series of Sn(g)(z) converge to g poi...
b) Consider the function g : [-π, π] → R, g(z) = -1 otherwise Does the Fourier series of Sn(g)(z) converge to g pointwise on [-π, π)? Provide evidence of your answer.
b) Consider the function g : [-π, π] → R, g(z) = -1 otherwise Does the Fourier series of Sn(g)(z) converge to g pointwise on [-π, π)? Provide evidence of your answer.
Consider the function y = x2 for x E (-7,7) . a) Show that the Fourier series of this function is...
Consider the function y = x2 for x E (-7,7) . a) Show that the Fourier series of this function is n cos(nz) . b) (i) Sketch the first three partial sums on (-π, π) (ii) Sketch the function to which the series converges to on R . c) Use your Fourier series to prove that 2and1)"+1T2 12 2 2 Tu . d) Find the complex form of the Fourier series of r2. . e) Use Parseval's theorem to prove...
Q#2 (22 points) (a) Find the Fourier series of the function by expanding the function as...
Q#2 (22 points) (a) Find the Fourier series of the function by expanding the function as an odd periodic function with a period of 10 units, as shown in Figure below. Plot the first, second, third and fourth partial sums of this Fourier series between -5 to +5 (Matlab is preferable). There will be single graph with 4 plots (b) Draw the amplitude versus frequency spectrum for first four non-zero terms of the Fourier series. Note that y(t) for -5<t<...
1 point) Consider the Fourier series: nTTc a. Find the Fourier coefficients for the function f(x) 1.2 an b. Use the computer to draw the Fourier series of f(a), for x E[-18, 18], showing clearly all...
1 point) Consider the Fourier series: nTTc a. Find the Fourier coefficients for the function f(x) 1.2 an b. Use the computer to draw the Fourier series of f(a), for x E[-18, 18], showing clearly all points of convergence. Also, show the graphs with the partial sums of the Fourier series using n5 and n20 terms. What do you observe?
1 point) Consider the Fourier series: nTTc a. Find the Fourier coefficients for the function f(x) 1.2 an b. Use...
a) Expand the given function in a Fourier series in the range of [-411, 41] (12...
a) Expand the given function in a Fourier series in the range of [-411, 41] (12 Marks) f(x) = { 1 0<x SI (sin(x) < x < 211 To what values will the Fourier Sine Series converge at x = -31, x = 0 and x = 27t? (3 Marks)
Question 4 (15 points): Fourier Series and its application 1. Find the Fourier series of the foll...
Question 4 (15 points): Fourier Series and its application 1. Find the Fourier series of the following function: 2. Use part(1) to show that (2k - 1)2 8 に1 Hint: Let x = π for the Fourier series of f(x) you found in part (1).
Question 4 (15 points): Fourier Series and its application 1. Find the Fourier series of the following function:
2. Use part(1) to show that (2k - 1)2 8 に1 Hint: Let x = π for...
find the Fourier cosine and sine series for the function f defined on an interval 0<t<L and sketch the graphs of the two extensions of f to which these two series converge: f(t)=1-t, 0<t<1...
find the Fourier cosine and sine series for the function f defined on an interval 0<t<L and sketch the graphs of the two extensions of f to which these two series converge: f(t)=1-t, 0<t<1
Problem 4. In class (or see notes on the homepage), we worked out the Fourier series for a square...
Problem 4. In class (or see notes on the homepage), we worked out the Fourier series for a square wave function. What do you get when you plug in z π/2? Does this seem! to make sense? (Warning: in general, the Fourier series for f(x) doesn't have to converge to the correct value of f(x)!)
Problem 4. In class (or see notes on the homepage), we worked out the Fourier series for a square wave function. What do you get...
2.4. HARMONIC FOURIER SERIES 57 Problem 2. Consider the function f in L? (0,2m) given by...
2.4. HARMONIC FOURIER SERIES 57 Problem 2. Consider the function f in L? (0,2m) given by f(t) = sin( 1.5) (when 0 < t < 2π Find the sine and cosine Fourier series expansion (3.1) for f. Choose a partial Fourier series approximation pn(t) for f (t). Then plot pn(t) and f(t) on the same graph. Compute the error llf - Pall. Does this Fourier series converge for t 2mj where j is an integer, and if so what does...
Fourier Series for Odd Functions Recall that if f is an odd function, f(-x)f(x). An odd Fourier series has only the...
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...
b) Consider the function g : [-π, π] → R, g(z) = -1 otherwise Does the Fourier series of Sn(g)(z) converge to g pointwise on [-π, π)? Provide evidence of your answer.
b) Consider the function g : [-π, π] → R, g(z) = -1 otherwise Does the Fourier series of Sn(g)(z) converge to g pointwise on [-π, π)? Provide evidence of your answer.
Consider the function y = x2 for x E (-7,7) . a) Show that the Fourier series of this function is n cos(nz) . b) (i) Sketch the first three partial sums on (-π, π) (ii) Sketch the function to which the series converges to on R . c) Use your Fourier series to prove that 2and1)"+1T2 12 2 2 Tu . d) Find the complex form of the Fourier series of r2. . e) Use Parseval's theorem to prove...
Q#2 (22 points) (a) Find the Fourier series of the function by expanding the function as an odd periodic function with a period of 10 units, as shown in Figure below. Plot the first, second, third and fourth partial sums of this Fourier series between -5 to +5 (Matlab is preferable). There will be single graph with 4 plots (b) Draw the amplitude versus frequency spectrum for first four non-zero terms of the Fourier series. Note that y(t) for -5<t<...
1 point) Consider the Fourier series: nTTc a. Find the Fourier coefficients for the function f(x) 1.2 an b. Use the computer to draw the Fourier series of f(a), for x E[-18, 18], showing clearly all points of convergence. Also, show the graphs with the partial sums of the Fourier series using n5 and n20 terms. What do you observe?
1 point) Consider the Fourier series: nTTc a. Find the Fourier coefficients for the function f(x) 1.2 an b. Use...
a) Expand the given function in a Fourier series in the range of [-411, 41] (12 Marks) f(x) = { 1 0<x SI (sin(x) < x < 211 To what values will the Fourier Sine Series converge at x = -31, x = 0 and x = 27t? (3 Marks)
Question 4 (15 points): Fourier Series and its application 1. Find the Fourier series of the following function: 2. Use part(1) to show that (2k - 1)2 8 に1 Hint: Let x = π for the Fourier series of f(x) you found in part (1).
Question 4 (15 points): Fourier Series and its application 1. Find the Fourier series of the following function:
2. Use part(1) to show that (2k - 1)2 8 に1 Hint: Let x = π for...
Problem 4. In class (or see notes on the homepage), we worked out the Fourier series for a square wave function. What do you get when you plug in z π/2? Does this seem! to make sense? (Warning: in general, the Fourier series for f(x) doesn't have to converge to the correct value of f(x)!)
Problem 4. In class (or see notes on the homepage), we worked out the Fourier series for a square wave function. What do you get...
2.4. HARMONIC FOURIER SERIES 57 Problem 2. Consider the function f in L? (0,2m) given by f(t) = sin( 1.5) (when 0 < t < 2π Find the sine and cosine Fourier series expansion (3.1) for f. Choose a partial Fourier series approximation pn(t) for f (t). Then plot pn(t) and f(t) on the same graph. Compute the error llf - Pall. Does this Fourier series converge for t 2mj where j is an integer, and if so what does...
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...
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