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3. Let f(x) be a function with period 27 such that f(x) = x in the interval - < x < T.
(i) Sketch a graph of f(x) in the interval -27 < x < 2π.
(ii) Show that the Fourier series for f(x) in the interval −ñ < x < π is:
1
f(x) = 2 (sin(x) = sin(2x)+sin(3x) - ...)
2
(Hint: you may need to do integration by parts, consider the limits that are being us
in any integral and whether the integrand is even or odd.)
(iii) By assigning an appropriate value for x, show that
ㅠ
4
1-
1 1
3
5
1
7
+
1
9
...

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4) [swHW] It turns out that any function that has a finite number of finite-magnitude discontinuities and a finite number of extrema (maximums and minimums) over a finite interval can be represented exactly with an infinite series of cosine and sine functions called a Fourier Series. The conditions that the function must meet, called the "Dirichlet conditions", are not very restrictive, so most functions you will encounter in physics will have an associated Fourier Series. To give you a sense of how this is possible consider a very simple f(x) = x function on the interval -π
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