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# Fourier Series and Fourier Transform

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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.

https://brainmass.com/math/fourier-analysis/fourier-series-fourier-transform-579494

#### 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

## 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;+=

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