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Fourier Analysis

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

Fourier Mathcad

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 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

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

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

Fourier Transform

Please see attachment. 1. What is the Fourier Transform for the convolution of sin(2t)*cos(2t). 2. Compute the inverse Fourier transform for X(w)= sin^2*3w 3. A continuous time signal x(t) has the Fourier transform X(w) = 1/jw+b where b is a constant. Determine the Fourier transform for v(t) = x*(5t-4)

Periodic Functions: Fourier Sine and Cosine Series

Please see attachment. #1 for following periodic functions acting on the given interval Do the following: a) Sketch 4 periods of the given function of period b) Expand the function in a sine - cosine Fourier Series f(x) = 2-x, -2<x<2 #2) Expand the function in a sine-cosine series and

Fourier Transforms and Wave Analysis

The question is the example on page 2 of the attachment (entitled 'Uniform Transducer'). it states that the center of the finger is at z'=L/4. I assume this is an arbitrary position. For Eq (2.4.6), the contribution from the left-hand finger is added. I'm not entirely sure how this equation is arrived at. It does not look like a

Exponential Fourier Series

Find the exponential Fourier series for x(t), y(t) and z(t). In each of three cases it is not necessary to do any integration. ω=2πf t = n/256 (t goes from 0 to 1 in increments of 1/256) x(t)= cos ω;t frequency= 2Hz y(t)= cos ωt frequency= 16Hz z(t)= the product of x(t) and y(t)

Fourier Integral and Convergence

Given the function below: Expand the function in an Fourier integral and determine what this integral converges to. f(x) = xe^-|4x| keywords: integration, integrates, integrals, integrating, double, triple, multiple

Fourier Series : Expansions and Differentiation

Let f (x) = |x| for x greater or equal to -1, less than or equal to +1 a) Write the Fourier series for f (x) on [-1,1]. b) Show that this series can be differentiated term by term to yield the Fourier expansion of f'(x) on [-1,1] c) Determine f'(x) and write it's Fourier series on [-1,1] d) Compare b and c. key

Examples of Fourier series and sums of numerical series.

We use the Fourier expansions of certain poynomial functions to compute the sum of some useful numerical series. The formulas are quite general and give, at the end, the Fourier expansion of every polynomial function. By the way, these formulas can be also used for a numerical approximation of pi=3.14....

Set of functions (Fourier Series and Signal Spaces)

Functions, Interval. See attached file for full problem description. Given the set of functions f1(t) = A1*exp(-t) f2(t) = A2*e^(-2t) Defined on the interval (0, infinity). (a) Find A1 such that f1(t) is normalized to unity on (0, infinity). Call this function PHI_1(t). (b) Find B such that PHI(t) and f2

Fourier Transform Integrals

Using the Fourier transform integral, find Fourier transforms of the following signals. xa(t) = t *exp(-αt) * u(t), α > 0; xb(t) = t2 * u(t) * u(1 – t) xc(t) = exp(-αt) * u(t) * u(1 – t), α > 0;

Fourier series, Fourier Transform and Partial Differential Equations

Please see the attached file for the fully formatted problems. ODE: 1. Solve ()'sinyxy=+. 2. Find the complete solution of the ODE ()()42212cosyyyx&#8722;&#8722;=. 3. Find the complete solution of the ODE ()46sinyy&#8722;=. 4. Find a second order ODE whose solution is a family of circle with arbitrary radius and center on t

Proof of sum of a given infinite series of constants (closed form).

The sum of the infinite series, 1/2^2 - 2/3^2 + 3/4^2 - 4/5^2 + ... is given as pi^2/12 - log 2 on pages 64-65 in the book "Summation of Series" by L. B. W. Jolley, 2nd ed., 1961, Dover Pubs. Inc. (the ^ symbol denotes exponentiation in the above series and sum). For most of the series in his book, he lists a source (referen

Derivation of Fourier Transform of a Gaussian

From equation 6 of the attached, derive equation 7. In the expression for s(v), the natural log does not apply to the term [(v-vo)/(dv/2)]^2 , should just be ln(2). Yes, the limits from minus infinity to plus inifinity are adequate. Thanks for the help!