### 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.

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.

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.

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

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. Please see the attached file for the fully formatted problems.

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

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)

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.)

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

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

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

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)

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

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).

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

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,....}.

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

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"

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.

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.

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

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

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

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)

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?

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

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

Please see the attached file for the fully formatted problems.

My problem is attached as jpg file.

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.