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How can I easily solve fourier series equations given below? I want to solve it quickly and easily in exams cuz I am not good with mathmatics very much..

For Example:

$a_n= \int_0^\pi e^{-t/2} \cos(nwt)\,dt$

Your help is appreciated alot.

Regards

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1 Answer 1

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The standard technique for that is partial integration, twice. You end up with an expression involving the original integral. Treat this as an equation to be solved for the integral. If you know the complex exponential function it is easier, since the integrand is the real part of $e^{(-1/2+inw)t}$, which is more easily integrated.

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  • $\begingroup$ yeah I did it that way but the text book states answer to be 0.504(2/1+16n^2) where as my answer is 2/pi (e^[-pi/2+i2 n pi]-e^[-pi/2-i2 n pi]) how can I solve it furthur $\endgroup$ Commented Oct 31, 2012 at 9:07
  • $\begingroup$ If you recall that $e^{x+iy}=e^x(cos y+i\sin y)$ (sometimes known as Euler's formula), you should be able to convert your answer to real form. $\endgroup$ Commented Oct 31, 2012 at 10:12
  • $\begingroup$ what is the answer if I replace cos with sine in the above given question $\endgroup$ Commented Oct 31, 2012 at 10:14

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