Finding c_n in Complex Fourier Series Expansion of $e^{i \omega t}$

In summary, the function f(t) = e^{i \omega t} is being expanded as a complex Fourier series with w not being an integer. The process involves joining exponentials, integrating and evaluating the limits, and using the identity for sinx. The final expression for the series is f(x) = \sum_{n=-\infty}^{n=+\infty} \frac{1}{\pi} \frac{1}{(\omega - n)} \frac{dw}{dt} sin(\pi \omega - \pi n) e^{int}.
  • #1
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I am expanding the function [tex]f(t) = e^{i \omega t}[/tex] from (-π,π) as a complex Fourier series where w is not an integer. I am stuck figuring out how the series expands with n.

[tex] c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{i \omega t} e^{-int} dt [/tex]

Join exponentials

[tex] c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{i t(\omega - n) [/tex]

Then integrate the function and evaluate the limits

[tex] c_n = \frac{1}{2 \pi} \frac{1}{i(\omega - n)} \frac{dw}{dt} (e^{i \pi (\omega - n)} - e^{ -i \pi(\omega - n)}) [/tex]

Use the identity for sinx

[tex] c_n = \frac{1}{\pi} \frac{1}{(\omega - n)} \frac{dw}{dt} sin(\pi \omega - \pi n)[/tex]

This is the point when I am not quite sure if I am done, or I need to do something more. Should I just put this into back into the expression for the complex Fourier series?

[tex] f(x) = \sum_{n=-\infty}^{n=+\infty} c_n e^{int}[/tex]
 
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  • #3
So then the function as a Fourier series would just be:

[tex] f(x) = \sum_{n=-\infty}^{n=+\infty} \frac{1}{\pi} \frac{1}{(\omega - n)} \frac{dw}{dt} sin(\pi \omega - \pi n) e^{int} [/tex]
 

1. How is c_n found in the complex Fourier series expansion of e^(iωt)?

To find c_n, we use the formula c_n = (1/T) * ∫(f(t)e^(-inωt)dt) where T is the period of the function and f(t) is the function being expanded.

2. What is the significance of c_n in the complex Fourier series expansion of e^(iωt)?

c_n represents the coefficient of the nth harmonic in the Fourier series expansion of e^(iωt). It determines the magnitude and phase of the corresponding harmonic component.

3. Can c_n be a complex number in the complex Fourier series expansion of e^(iωt)?

Yes, c_n can be a complex number. This indicates that the corresponding harmonic component has both magnitude and phase, and is essential for accurately representing complex periodic functions.

4. How does changing the value of n affect c_n in the complex Fourier series expansion of e^(iωt)?

As n increases, c_n represents higher frequency harmonic components in the Fourier series expansion. This means that the function will have a more accurate representation with a larger number of harmonics, but the complexity and number of calculations required will also increase.

5. Is c_n the only factor that determines the accuracy of the complex Fourier series expansion of e^(iωt)?

No, c_n is only one factor that contributes to the accuracy of the Fourier series expansion. The choice of the period T and the function f(t) also play a crucial role in determining the accuracy of the expansion.

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