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Fourier Analysis

Proof of sum of a given infinite series of constants (closed form).

The sum of the infinite series, 1/2^2 - 2/3^2 + 3/4^2 - 4/5^2 + ... is given as pi^2/12 - log 2 on pages 64-65 in the book "Summation of Series" by L. B. W. Jolley, 2nd ed., 1961, Dover Pubs. Inc. (the ^ symbol denotes exponentiation in the above series and sum). For most of the series in his book, he lists a source (referen

Fourier coefficients outputs

Fourier coefficients / b1, b2, b3, b4, b5... b11. -------------------------------------------------------------------------------- I have an output of an electronic device (full wave rectifier) that gives a sine wave with the negative part transposed symmetric to xx so that the function is always positive. I have to find the f

Matlab plots of the FFT of sequence

(See attached file for full problem description) For sequence x[n]=[1 1 1 1 0 0 0 0] for n=0:7, so N=8 Using above x[n]: a) stem(x); b) Use the shift theorm to plot x delayed by 1, 4, 5, 6, and 8 samples, and plot the result for each. Remember the shift theorem says a delay by t0 seconds is equal to multiplying the spe

Discrete time Fourier transform of sequence and Matlab plot

Please see the attached file for full description. Calculate by hand the X(omega), DTFT of the sequence x[n]=[1 1 1 1 0 0 0 0] for n=0:7, zero else. Using Matlab, plot the real and imaginary components of your result for X(omega) for omega=0:0.01:2*pi, one plot for the real, one part for the imaginary. On the same plot

Fourier series

Find the Fourier series in trigonometric form for f(t) = |sin(pi*t)|. Graph its power spectrum.

Bessel and Legendre Series : Fourier-Legendre Expansions

8. The first three Legendre polynomials are P0(x) = 1, P1(x) = x, and P2(x) = 1/2(3x2- 1). If x = cosθ , then P0( cosθ ) = 1 and P1( cosθ ) = cos θ . Show that P2( cosθ ) = 1/4( 3cos2θ + 1 ). 9. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (θ) = )of F(

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

Sturm-Liouville expansion

Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer. 3. Expand... in terms of the eigenfunctions of the Sturm-Liouville problem. Please see attached.

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

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.

Fourier Series Sketched

f(x)= {0 -2<x<0 f(x+4) = f(x) {1 0<x<2 I have to find the Fourier series of the problem and sketch the graph of the function at 3 periods. I'm not sure if you need my text and what sections were covering or not so I'll just give it. "Elementary differential equations and boundary value prob

Gibbs Phenomenon and Fourier Series expansion

4. In this problem, you will devise a computer experiment to investigate Gibb's phenomenon, which is the presence of spurious oscillations in the graph of a truncated Fourier series near the places where the full Fourier series is discontinous. Choose any function you like that demonstrates Gibb's phenomenon. Your goal is to

Fourier Transform and Schrodinger Equation

Please view the attached file for the full description of the two questions being analyzed. Essentially, this posting is asking the following: Solve the Schrodinger equation with different potentials using the Fourier transform.

Finding a Fourier Series

Suppose f(t) and g(t) are 2&#960; periodic functions with Fourier series representations {see attachment}. Find the Fourier series of {see attachment}.

Fourier Transforms

Use the Fourier transform to solve the one-dimensional wave equation. See attached file for full problem description.

Fourier Transform of the Signal

If the Fourier transform of the signal v(t) is v(w) = AT sinwt / wt then the energy contained in v(t) is a) (A^2)/2 b) A^2 c) (A^2)T d) (A^2)T/2

Complex Fourier Series

Please see the attached file for the fully formatted problem. Let Y: R --> R be the periodic function whose restriction to [0,1] is X (0,1/2) - X(1/2,1) Y is an odd function. S 1--> 0 Y(x) cos 2pi kx dx = S 1/2-->-1/2 Y(x) cos 2pi kx dx Vk Conclude the the complex Fourier Series...can be expressed in the form...

Eignevalues and Eigenvectors of the Fourier Transform

Problem attached. "Eigenvalues and Eigenvectors of the Fourier Transform" Recall that the Fourier transform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself. Let .. be an element of L2(?cc,cc). Let..= , the Fourier transform of.., be defined by ..... It is clear that ..... are square-integrable fu

Inner Product : Fourier Transform

Please see the attached file for the fully formatted problem. 2. Let p be a fixed and given square integrable function, i.e. 0 < S g(x)g(x) dx = ||g^2|| < inf The function g must vanish as |x| ---> inf. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x

Fourier analysis

(a) Explain the relationship between the spectral components indicated above and the corresponding graph showing the motion of the surface of the motor plotted as a function of time. What do they represent? (b) Given the amplitudes indicated in the above diagram, copy and complete the table below for the amplitude of the vib