Fourier Series Solved Problems

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I tried using Euler's Identity to separate them from each other but then I am stuck on: $$f(t)=\left(\frac{e^...

I was trying to fit some data to a Fourier series expansion, and ended up with the following: Why is there no improvement near the gap when adding more terms to the expansion? This is the equation for Fourier integral Now isn't x=t?

\[ = \frac\int_^ = \frac\int_^ = L\] \[\begin &= \frac\int_^ = \frac\int_^\\ & = \frac\left.

\right|_^L\\ & = \frac\left( \right)\left( \right) = 0\hspace\hspace\hspace\hspacen = 1,2,3, \ldots \end\] \[\begin &= \frac\int_^ = \frac\int_^\\ & = \frac\left.

It is instead done so that we can note that we did this integral back in the Fourier sine series section and so don’t need to redo it in this section.

Using the previous result we get, \[ = \frac\hspace\hspacen = 1,2,3, \ldots \] In this case the Fourier series is, \[f\left( x \right) = \sum\limits_^\infty \sum\limits_^\infty = \sum\limits_^\infty \] If you go back and take a look at Example 1 in the Fourier sine series section, the same example we used to get the integral out of, you will see that in that example we were finding the Fourier sine series for \(f\left( x \right) = x\) on \( - L \le x \le L\).

\[\begin& \int_^ = \left\{ \right.\ & \ & \int_^ = \left\{ \right.\ & \ & \int_^ = 0\end\] So, let’s start off by multiplying both sides of the series above by \(\cos \left( \right)\) and integrating from –\(L\) to \(L\).

Doing this gives, \[\int_^ = \int_^ \int_^\,dx\] Now, just as we’ve been able to do in the last two sections we can interchange the integral and the summation.

\[\begin & = \frac\int_^ = \frac\left[ \right]\ & = \frac\left[ \right] = \frac\left[ \right] = L\end\] \[\begin = \frac\int_^ & = \frac\left[ \right]\ & = \frac\left[ \right]\end\] At this point it will probably be easier to do each of these individually. \right|_^0 = \frac\sin \left( \right) = 0\] \[\begin\int_^ & = \left.

\right|_0^L\ & = \left( \right)\left( \right)\ & = \left( \right)\left( \right)\end\] So, if we put all of this together we have, \[\begin & = \frac\int_^ = \frac\left[ \right]\ & = \frac\left( \right)\,\,\,,\hspace\hspacen = 1,2,3, \ldots \end\] So, we’ve gotten the coefficients for the cosines taken care of and now we need to take care of the coefficients for the sines.

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