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Fourier Analysis

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

Convolution of Fourier transforms : Associativity Law

Please see the attached file for the fully formatted problems. The convolution of two functions f and g is defined to be the new function f * g. whenever the integral converges. (a)Is it true that f*g = g*f? Why? (b) If F(k) = (i//) f°° exp (?ikx) f(x) dx and G(k) = (i//) f°° exp (?ikx) g(x) dx are the Fourier tra

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