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# Fourier Analysis

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### Real Analysis : Fourier Transform for L1R

Proof regarding a Fourier Transform for L1R. Show 1/2pi ∫-1 -->1 (1 -|y|)e^(iyx) dy = 1/2pi [(sin x/2)/(x/2)]^2 Please see the attached file.

### Fourier Series and Convergence

Let f(x) ={0 -pi<x<0 {x 0<x<pi a) Compute the Fourier series for f on the interval [-pi, pi] b) Sketch the function to which the series converges. Please see the attached file for the fully formatted problems.

### 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.

### Fourier Sine and Cosine Series, Laplace's Equation and Wave Equation

Please assist me with the attached Fourier Series problems. Please see the attached file for the fully formatted problems.

### Express Fourier series in sine-cosine form and complex form and find convergence.

Express Fourier series in sine-cosine form and complex form and find convergence. 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)

### Fourier Series of a Periodic Extension

5. Find the Fourier series of the periodic extension of the function... (see attached) Please do Question 5 only. Show step by step work and explanation of the solution. (Answer is provided in the attachment.)

### Fourier Cosine Series

Please do section B, problem 4. Show step by step work and explanation of the solution. (Answer is provided in the attachment.)

### Odd Fourier Extension

Please give step by step work and explanation of solution. (Solution is provided in the Attachment.)

### Fourier Series of the Periodic Function

The answer is provided - please provide step by step work and explaination and solution. 4. Find the Fourier series of the periodic function attached...

### 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)

### 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

### Application of Fourier Transfors to Diffusion

Using Fourier Transforms, solve the one-dimensional equation for a point source located at x=xo, i.e., at time zero, c(x,0) = (delta)(x-xo).

### 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

### Fourier Series Expansions

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,....}.

### 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

### Forcing Function and Decay

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.

### Evaluate: Complex Fourier Transform

Use the Fourier transform to solve the following differential equation: g" + 2g' + 5g = delta(x) Where delta is the dirac delta function (impulse). Please view the attached file for the full problem description.

### 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}.

### Inverse Fourier Transform Complex

Calculate the inverse Fourier transform of 1/(w^2+2iw-2) in two ways: using the definition and using partial fractions.

### Fourier Transforms

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

### Fourier transforms and residual 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

### Fourier Series for Periodic Functions

Find the Fourier series as well as the first three partial sums of the Fourier series. F(x)=x-x^2 if -1<x<1

### To What Function does the Fourier Series of a Give Function Converge?

Please see the attached file for the fully formatted problems.

### Fourier Series - Vector Space

My problem is attached as jpg file.

### Fourier Series - Uniform and Pointwise Convergence Problem

Please see the attached file for the f(x) function. a) On the interval [a,b], does the sequence of functions converge pointwise? If yes, what is the limit function? Is the convergence uniform? b) Answer the same three questions, but now let the function be defined on the real line.