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Fourier Analysis

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

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 Cosine Series and Fourier Series Expansion

In the interval (-pi, pi), &#948;n(x) = (n/x^1/2) e ^(-n^2 x^2) a) Expand &#948;n(x) as a Fourier Cosine Series. b) Show that your Fourier Series agrees with a Fourier expansion of &#948;n(x) in the limit as n--> infinity. Please see the attached file for the fully formatted problems.

Signal Function : Fourier Transform and Coefficients

A signal function f(t) of period 2 pi is given by: (See attached file) As required by my question I have drawn the above signal in the interval -4pi < t < 4pi which I beleive to be a sawtooth signal. I also need to find if f(t) is odd, even or neither, hence state which coefficients, if any, are zero. If there

Fourier Transformation : Even Function

A function h(x) is positive or zero for all values of x. Assume h(x) is even. If the Fourier transformation of h(x) is H(u) show that... (See attachment for full question)

Sum of Series using Fourier Series

Use the Fourier series of the function f(x) and the properties of Fourier series to obtain the Fourier series of... (aee attachment for full question)

Note to YINON

Yinon, If you do not wish to work on the Laplace diffusion problem anymore, I would like to see the solution you have formulated thus far and will pay you some credits even though it is not entirely correct. Please leave me a note. Thank you!

Laplace/Fourier Transforms

Please see the attached file for the fully formatted problems. The problem is number 8 on page 989 of the 2nd Edition of Greenberg's Advanced Engineering Mathematics book, PART C ONLY. This is in section 18.4 (Chapter 18 is the Diffusion Equation), in the exercises at the end of the section. PLEASE NOTE: I have scanned th

Fourier Series

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

Fourier Series

1. Find the Fourier series expansions of f(x) = Co + C1x^2 with respect to the following two orthonormal bases on the interval [0,L] (L>0) a) {(1/L)^.5, (2/L)^.5*cos*((k*pi*x)/L)|k=1,2,...} b) {(2/L)^.5*sin*((k*pi*x)/L)|k=1,2,....}

Forcing Function

What is the solution to Y''(x) + 2y'(x) + 5y(x) = f(x) Where f(x) is a given forcing function, and y and f both decay to 0 as x ---> + or - INF Note: should read "as x approaches plus or minus infinity"

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.

Solve using Fourier transforms and residue calculus

3. The nefarious Evil Corp is dumping radioactive pollutant into a river moving with speed c. Define x to be the downstream coordinate, with the pollutant being injected at x=0... (see attachment for rest of question)

Fourier Series Proof

Please see the attached file for full problem description. Q = e^ (i n pi alpha) + e^ (-i n pi alpha) and Q = 0. How can Q be two different things?

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

Fourier Transform : Equivalent Width

"Equivalent Widths" Suppose we define for a square-integrable function f(t) and its Fourier transform ..... the equivalent width as .... and the equivalent Fourier width as .... a) Show that ..... is independent of the function f, and determine the value of this const. b) Determine the equivalent width and the equiva