Cn coefficient for a Fourier series

In summary, the conversation discusses determining the Cn coefficient for a Fourier series and the confusion surrounding the substitution and calculation of limits. The conversation also points out a misprint and provides a solution using the identity e^ix + e^-ix = (cosx)/2.
  • #1
sekowe
2
0

Homework Statement



I have an example were they determine the Cn coefficient for a Fourier series:
2dcisrb.png


My problem is i don't follow what happens in the following moment:
25jdr47.png


neither am i sure how the two e^-inwt comes to be cos(nwt) in the last part.

Homework Equations





The Attempt at a Solution



The only solution I can come up with is to reverse the limits what is obvious and then say: t=-a --> -1*t=-1*-a and then change a for -a in the limit and t for -t. However this doesn't work in other similar problems so I don't think think this is right.
 
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  • #2
welcome to pf!

hi sekowe! welcome to pf! :smile:

there's a misprint :redface:

the first exponent in the third line should be positive

between the second and third lines, they've substituted u = -t, du = -dt, and then renamed u as t again!

and of course the corrected third line let's you use eix + e-ix = (cosx)/2 :wink:
 
  • #3
A misprint... Yeah that clears up a lot :D

Thank you very much!
 

1. What is the Cn coefficient for a Fourier series?

The Cn coefficient, also known as the Fourier coefficient, is a complex number that represents the amplitude and phase of a specific harmonic in a Fourier series. It is used to decompose a periodic function into a series of sinusoidal functions.

2. How is the Cn coefficient calculated for a Fourier series?

The Cn coefficient is calculated using the formula: Cn = (1/2π) ∫f(x)e^(-inx)dx, where f(x) is the periodic function and n is the harmonic number. This integral is evaluated over one period of the function.

3. What is the significance of the Cn coefficient in a Fourier series?

The Cn coefficient is significant because it determines the contribution of each harmonic to the overall representation of the periodic function. The magnitude of the coefficient represents the amplitude of the harmonic, while the phase determines the position of the harmonic in relation to the other harmonics.

4. How do the values of the Cn coefficient affect the shape of a Fourier series?

The values of the Cn coefficient determine the shape of a Fourier series by determining the amplitude and phase of each harmonic. The more harmonics included in the series, the closer the representation will be to the original function.

5. Can the Cn coefficient be negative?

Yes, the Cn coefficient can be negative. Negative coefficients represent a phase shift of π radians, or 180 degrees, for the corresponding harmonic. This can result in a mirrored or inverted representation of the periodic function.

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