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# Fourier Analysis

### Fourier transform in distribution sense

Please solve problem number 1 on the attached PDF file

### Dirichlet's Test for Convergence of a Series

Verify that (1/2i)sum {(e^(inx))/n} (where n does not equal zero in the sum) is the Fourier series of the 2 Pi periodic function defined by f(0)=0 and f(x)=-Pi/2-x/2 for between -pi and 0 and f(x)= Pi/2-x/2 for x between 0 and Pi. Show that the Fourier series is convergent. (Note that the function is not continuous. Show t

### Inverse fourier transform of a bessel function

The equation is J1(wg) where J1 is bessel functon of the first kind of order 1, w is angular frequency and g is a constant. How do I inverse fourier transform this equation to time domain representation? Thanks

### Lebesque measure in R^2

How would the construction of the Lebesgue measure in R2 change if we assume at the beginning that the measure of the unit square [0, 1]Ã?[0, 1] is =2? What would the measure of a circle of radius 1 be? Why?

### F(x)=2x question

Determine the fourier series for the function:- F(x) = 2x in the range -pi to pi and are periodic the the period 2pi Sketch both the expected and fourier series graphs and use mathcad to check

Sketch graphs using Mathcad for the following functions and state whether they are odd or even or neither y3= In x { y4= x sin x { in the range -pi to pi and are periodic of period 2pi y5= x {

### expected Fourier series graphs

Determine the Fourier series for the function F(x)= { x when 0 <x<Pi . { {0 when Pi< x <2Pi and is periodic of the period 2Pi. Sketch both the expected Fourier series graphs using Mathcad to check solution.

### Fourier analysis and Rayleigh quotient

I have posted a total of four questions as well as the "Rayleigh Quotient" that is referenced in problem # 4 part c. I thank you for your time and consideration in these matters. Please let me know if any further clarification is needed because I can provide notes that would help with solving these problems.

### PDE solutions using Fourier Transforms

See attached PDF for the compiled LaTex. I would prefer any solution include the Latex Source, however, if you are unable to use LaTex, you may use another format. Exercise 1 Solve, using Fourier Transforms frac{partial^{2}u}{partial x^{2}}+frac{partial^{2}u}{partial y^{2}}=0 for 0<y<H, -infty<x<infty subject to the i

### Fourier Series Expansion: Example Problem

Please see the attached file for the fully formatted problems. See f(x) in the file. 1. Sketch f(x) over -3<x<3 2. Is f(x) odd, even or neither? 3. Solve for the Fourier coefficients. 4. Write out the Fourier series expansion up to n=3

### Fourier Series Expansion

Please see the attached file for the fully formatted problems. Define periodic p=2 function as follows: f(x) = -exp(x) -1<x<0 exp(x) 0<x<1 Solve for the Fourier coefficients

### Find the 7 Fourier coefficients of the function...

There is an attached file with further information regarding the problem. Find by inspection the first seven Fourier coefficients {a0, a1, b1, a2, b2, a3, b3} of the function: f(x) = 14-cos(Pi*x/10) + 3sin(Pi*x/10) + 0.5cos(Pi*x/5) + 5sin(3*Pi*x/10)

### Fourier Series and Gibbs Phenomenon

Please see the attached file for the fully formatted problems. Consider the ODE y" - a^2 * y = H(x-Pi/2) y(0)=y(Pi)=0 0<x<Pi Where H is the step function. solve for y using Fourier series The Fourier series for the step function exhibits the Gibss phenomenon. Will the solution y(x) exhibit it too? explain why o

### Fourier Transforms and Convolution Theorem

Consider the ODE ..... with the boundary conditions y(x) bounded as.... Assume that b is real and positive and that g(x) behaves in such a way so that a bounded solution is possible. (a) Compute the Fourier transform of the solution (b) Use the convolution theorem to solve the ODE and express the solution as an integral i

### Fourier Series Continuous Function Intervals

See attachment.....please show each step in detail in the solution. Let f(theta) be a continuous function on the interval...and let fn(theta)denote its nth Fourier series approximant....

### Fourier transform

Please see the attached file and include an explanation of problem. Thank you. 1. Compute the Fourier transform for x(t) = texp(-t)u(t) 2. The linearity property of the Fourier transform is defined as: 3. Determine the exponential Fourier series for: 4. Using complex notation, combine the expressions to form a singl

### Example of Developing a Fourier Series

Develop in series of Fourier the periodical function with period 2pi (see attached).

### Fourier Series Analysis

Please assist with the attached file. The function f(x) has a period of 2pi. It is defined on the interval [-pi, pi] by...

### How to Find Normal Modes for a Wave Equation

Could you please show me how to do the problem attached? You don't have to do the first part (proving solutions to the wave equation by a separation of variables) as I know how to do that. Please start where it asks what is a normal mode, etc. See the attached file.

### Fourier transforms

1. Find the Fourier sine series of f(x)=1, 0<x<L/2. Use this to prove that 1-1/3+1/5-1/7+...=Pi/4 2. Solve df/dt=d^2 f/dx^2 - f subject to the initial condition f(0,x)=1 if |x|<L/2 or f(0,x)=0 if |x|>L/2 Please see the attachment to view the questions with correct mathematical notation (and also phrased slightly differe

### The Fourier coefficients of a derivative

Let f be a 2 pi periodic, differentiable function with Fourier coefficients a_n and b_n. Let (a_n)*, (b_n)* be the Fourier coefficients of f'. a) Show that (a_0)*=0 b) Use integration by parts to find a formula for the Fourier coefficients of f' in terms of the Fourier coefficients of f. (The attachment contains the

### Waveforms : Fourier and Laplace Transforms

Have the following question regarding extracting information from a waveform a. Write down an expression, in the time domain, for the signal in the diagram above. b.Derive the Laplace transform for this signal. c.Use Laplace transform analysis to derive the Fourier transform in its simplest form. All details in the a

### Heat Problem on the Circle

So I am able to get to u(x,t) = sum_{n=-infty}^{infty} Cexp(iBx - ktB^2) + Dexp(-iBx - ktB^2) because the general series is sum_{n=-infty}^{infty} X(x)T(t) You have 1 coefficient in your general series, yet I have 2. How do I get it into 1?

### Sine to cosine to exponents transition in Fourier analysis.

Could one please explain the sin-cos to exponents transition in Fourier analysis?

### Heat Equation on Circle

Solve the heat problem on the circle u_t = ku_{xx} u(x,0) = phi(x) where phi(x) is the 2l periodic extension of phi using the separation of variables. I am able to go as far as u = XT -X''/X = lambda where lambda = beta^2 usually the solution for X'' + beta^2 * X = 0 is Ccos(beta * L) + D sin

### Fourier Analysis and Series : Periodic Functions and Convolution

Please see the attached file for the fully formatted problems.

### Cosine Fourier Transforms

In the two problems below find the Fourier Cosine Transform of the given f(x) and write f(x) as a Fourier integral. Generate the transform and fourier integral using the Cosine Transform..please show all steps and compare the answer to the previous problem 2x+2a -a<x<0 f(x) = -2x+

### Exponential Fourier Transforms and Fourier Integrals

Please see the attached file for the fully formatted problems. In the two problems below find the exponential Fourier Transform of the given f(x) and write f(x) as a Fourier integral. 2) 2x+2a -a<x<0 f(x) = -2x+2a 0<x<a 0 |x|>a The Fourier is of the type (see attachme