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# Fourier Analysis

### Least-Squares Approximation and Mean Square Error

5. (a). Find the least squares approximation of sin(&#960;x) over the interval [-1,1] by a polynomial of the form ao+ a1x+a2x². (b). Find the mean square error of the approximation. Note: Part (a) that is suppose to be sin(piex) and the polynomial is a sub 0, a sub 1, a sub 2. For some reason it wouldn't allow me t

### Fourier Series : Least-Squares Approximation and Mean Square Error

4. (a). Find the least squares approximation of e^x over the interval [0,1] by a polynomial of the form ao+a1x. (b). Find the mean square error of the approximation.

### Forurier Series and Coefficients

1.(a) Evaluate the Fourier coefficients ao, an, bn and the exponential coefficient Dn for the waveform shown below; do not use MATLAB or a calculator for integrations. (b) Plot 2 or 3 cycles of the Fourier series using MATLAB and verify whether the plot matches the given waveform. (c) Find C0 and Cn from the answer to part (a)

### One Dimensional Heat Equation on a Finite Rod

Can someone please solve this heat equations with details on how to arrive at the solution.? Solve the heat equation: u_t = 3*u_xx with the following I.C/B.C: u(0,t) = u(L,t) = 0 u(x,0)=L*[1-cos(2*Pi*x/L)]

### Use Parseval's equality

We are using the book Methods of Real Analysis by Richard R. Goldberg (See attached file for full problem description) --- 12.5-2 Show that the Fourier series for is a) Use 12.5E to show that Fourier series at t=0 converges to . Deduce that 12.5E: Theorem. Let ( this

### Sine and Cosine Fourier Series and Determine the Sum of Each

Write the cosine and sine Fourier Series. Determine the sum of each. { 4x 0&#8804;x&#8804;2 f(x) ={ -3 2<x<4 { 1 4&#8804;x&#8804;7 Please see the attached file for the fully formatted problems.

### Find Fourier Series

Find Fourier series for f(x) = {-4 for x greater than/= -pi, and x less than/= 0 { 4 for x greater than/= 0, and less than/= pi

### Fourier Series and Fourier Sine and Cosine Series

1.) Find fourier series of f(x)=4, x greater than -3 and less than 3 and 2.) Find fourier series of f(x) = x^2-x+3, x greater than -2 and less than 2 and 3.) Write the cosine and sine fourier series f(x)=x^2 for x greater than 0 and less than 2

### Find Fourier Series and Cosine and Sine Fourier Series

1.) Find fourier series of f(x)=4, x greater than -3 and less than 3 and 2.) Find fourier series of f(x) = x^2-x+3, x greater than -2 and less than 2 and 3.) Write the cosine and sine fourier series f(x)=x^2 for x greater than 0 and less than 2

### Fourier Series Expansions : Periodic Functions, Graphing and Finding Terms

Find the first four terms of the Fourier series expansion of the function f(x) = (1 - (x^2/pi^2))per on the interval (-pi, pi). Make a graph of f(x) with the two Fourier series expansions. Show three periods of f(x).

### Fourier Transform Direct Computation

See attachment. Please use the Fourier transform tables rather than a direct computation. Cheers.

### Fourier Coefficient of the Signal

Find the nth Fourier coefficient (in terms of the an's) of the signal y(t) defined by: y(t) = x(t+1) + x(1-t), for all t in <R>. Please see attachment for more readable form of question.

### Finding Values Using Dirichlet's Theorem

For of the periodic functions , I need to find the value to which the Fourier series converges at x= 0 , Pi/2 , - Pi/2 , Pi , -Pi , 2Pi , - 2Pi Using Dirichlet's theorem (See attached files for full problem description).

### Fourier Series

Please help ... what is the Fourier series expansion of f(t). It is a multiple answer question. (See attached file for full problem description)

### Fourier Cosine Transform

The problem is from Fourier Cosine and Sine Transforms, and Passage from Fourier Integral to Laplace Transform: Solve using a cosine or sine transform. u'' - 9u =50e^-3x (0<x<infinty) u'(0) = 0, u(infinity)bounded

### Passage from Fourier Integral to Laplace Transform

Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer. Fourier Cosine and Sine Transforms, and Passage from Fourier Integral to Laplace Transform: Given the rectangular pulse.. Please see attached.

### Calculate Fourier Transforms Using a Table

Evaluate the following using Appendix D (Fourier Tranform Table). You may need to use more than one entry. Cite, by number, any entrie that you use. a)F{4x^2e^(-3|x|)} c) F{cos 3x/(x^2 + 2)} Please see the attached file for the fully formatted problems.

### Fourier integral representations

Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer. Derive the Fourier integral representations of the following. At which points, if any, does the Fourier integral fail to converge to f(x)? To what values does the integral converge at those points? (b) f(x) = {x

### Eigenfunctions - Expand the function

Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer. Expand the function f(x) = {x^4, 0 <= x < 2 {0, 2 <= x <= pi in terms of the eigenfunctions of the given eigenvalue problem. Use computer software, such as the Maple int command, to evaluate the expan

### Sturm Liouville problem

4. Use the results of Exercise 3 to recast each of the following differential equations in the Sturm-Liouville form (1a). Identify p(x), q(x), and w(x). (a) xy" + 5y' + lambda xy = 0 (b) y" + 2y' + xy + lambda x^2y = 0 (c) y" + y' + lambda y = 0 (d) y" - y' + lambda xy = 0 (e) x^2y" + xy' + lambda x^2y = 0 (f) y" + (cot

### Eigenvalues and Eigenfunctions

1. Identify p(x), q(x), w(x) alpha, beta, gamma, delta, solve for the eigenvalues and eigenfunctions, and work out the eigenfunction expansion of the given function f. If the characteristic equation is too difficult to solve analytically, state that and proceed with the rest of the problem as though the lambda_n's were known.

### Partial Sums of Fourier Series

The problems are from Fourier Series, Fourier Integral, and Fourier Transform. Please show each step of your solution. Obtain a computer plot of the partial sums of the Fourier series (19) of the periodic function shown in the attached figure, for (a) n =2 (b) n = 5 (c) n = 10 (d) n = 20 (e) n = 30 (f) n = 50 Ple

### Fourier Series of a Periodic Fuction: Convergence

Work out the Fourier series of f, given over one period as follows. At which values of x does the series fail to converge to f(x)? To what values does it converge at those points? (h) |cos x| for all x (k) x on 0<x<1, 1<x<2. See the attached file.

### Fourier Transform

Find the Fourier transfrom of the following function: f(t) = te^(-2t), for t > 0

### Fourier Series help

Find the complex Fourier series coefficients for the function x(t) depicted in the below figure Please see

### Fourier Coefficients

Here, we have to find f(t) from the given value of Cn. I am not able to arrive at f(t)={3/[5-4cos(pit+pi/20)} despite many attempts. Please show me how to arrive at the final expected f(t) value.

### Determine the complex Fourier series coefficients

Determine the complex Fourier series coefficients for the following function Please see attached.

### A problem on Contour Integration, part of Fourier Lessons

The following problem is a portion of the proof about Fourier Transforms for... Please see attached.