Explore BrainMass
Share

Fourier Series and Fourier Transform

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

Please show all steps.

1. Let f(x) be a 2pi- periodic function such that f(x) = x^2 −x for x ∈ [−pi,pi].
Find the Fourier series for f(x).

2. Let f(x) be a 2pi- periodic function such that f(x) = x^2 for x ∈ [−1,1]. Using
the complex form, find the Fourier series of the function f(x).

3. See attachment for better formula representation.
a. Verify that the function g satisfies the condition ∫ |g(x)| ^2 dx < ∞
b. Compute the Fourier Integral of g(x).
c. Determine what the Fourier Integral of g(x) converges to at each real
number.

4. Consider the Gaussian function : (see in attachment)
a. Sketch the graph in EXCEL of the Gaussian function when a = −0.1,
a = 1 and a = 10 in the same frame.
b. Compute the Fourier Transform of the Gaussian function for a = 1.

© BrainMass Inc. brainmass.com October 25, 2018, 9:35 am ad1c9bdddf
https://brainmass.com/math/fourier-analysis/fourier-series-fourier-transform-579494

Attachments

Solution Preview

The solution is attached below in two files. the files are identical ...

Solution Summary

The solution shows, step by step, how to calculate the Fourier series and Fourier transforms of functions (including a full derivation of the transform of a Gaussian).

$2.19
See Also This Related BrainMass Solution

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 the x-axis, i.e., ()22xayb&#8722;+= where a and b are arbitrary constants.
Fourier series, Fourier Transform and Partial differential equation
5. Write the Fourier series for ()cosftt=.
6. Find the Fourier series of a periodic signal with ()()exp,11ftt=&#8722;&#8722;<
7. Find the (two - side) Fourier transform ()(){}FWFft= of ()()expfttt=&#8722;.
8. Find the Fourier transform ()Xf of ()()()expcos2cxtt&#960;=&#8722; .
9. Solve the partial differential equation xtxzzx+= for (),,0,0zxtxt&#8805;&#8805; with the condition and (),00zx=()0,0zt= Hint : Use Laplace transform
10. Solve for (,zxt the partial differential equation
,0,01xxtzzztx=+&#8805;&#8804;&#8804;
with the conditions ()()0,1,0xxztzt= for all t and ()2,02sinzxx&#960;= for all x.
Laplace and Inverse Laplace Transform
11. Find the laplace transform of ()()2sintftetut&#8722;=.
12. Find the inverse laplace transform of ()24212sseeFSs&#8722;&#8722;&#8722;+=.
13. Find the inverse laplace transform of ()()()()2224211sFSss+=++.
Eigenvalue and Eigenvector
14. Find the eigenvalue and eigenvector of 112121011A&#8722;&#9121;&#9124;&#9122;&#9125;=&#8722;&#9122;&#9125;&#9122;&#9125;&#8722;&#9123;&#9126; and . 111111111B&#9121;&#9124;&#9122;&#9125;=&#9122;&#9125;&#9122;&#9125;&#9123;&#9126;
Vector space, Basis, Dimensions
15. Find condition on so that ,,abc()3,,abcR&#8712;belongs to the space generated by and ()()2,1,0,1,1,2,uv==&#8722;()0,3,4w=&#8722;.
16. Let W be the subspace of 4R generated by the vectors ()()1,2,5,3,2,3,1,4&#8722;&#8722;&#8722; and ()3,8,3,
a. Find a basis and the dimension of W.
b. Extend the basis of W to a basis of the whole space 4R.
17. Let Uand W be subspaces of 5R such that
U is spanned by ()()(){}1,3,3,1,4,1,4,1,2,2,2,9,0,5,2&#8722;&#8722;&#8722;&#8722;&#8722;&#8722;&#8722;&#8722;
W is spanned by ()()(){}1,6,2,2,3,2,8,1,6,5,1,3,1,5,6&#8722;&#8722;&#8722;&#8722;&#8722;&#8722;
a. Find the basis of ()UW&#8745;.
b. Find dim ()UW+ and dim ()UW&#8745;.
Residues
18. Evaluate 220sin53cosd&#960;&#952;&#952;&#952;+&#8747;.
19. Evaluate ()32122cdzzzz++&#8747;&#56256;&#56438; where c is the counter - clockwise.
20. Evaluate 2322146zdzzzz&#8734;&#8722;&#8734;&#8722;&#8722;&#8722;&#8722;&#8747;.
System of linear equation
21. Find the value of so that the solution of the following equations exists. By using that value of solve those equations. k,k
12312312323472311xxxxxxxxx&#8722;+=&#8722;++=&#8722;&#8722;&#8722;+=

View Full Posting Details