- #1
jellicorse
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I have been trying to follow how the complex Fourier coefficients are obtained; the reference I am using is at www.thefouriertransform.com. However I am unable to follow the author's working exactly and wondered if anyone could help me see where I am going wrong.
First, I understand that the coefficient c_n=\frac{1}{T}\int^T_0 f(t) e^{\frac{-i2\pi nt}{T}}dt
Given a square wave, of period 2T, and amplitude :
http://s90.photobucket.com/user/jonnburton/media/SqareWave_zpsbdf193c1.jpg.html
It can be seen directly from the graph that c_0=\frac{1}{2} (I also managed to calculate this, too.)
But I can't follow how the author obtained expressions for c_n.
This is what I have done in attempt to obtain c_n:
c_n=\frac{1}{T}\int^{\frac{T}{2}}_0 1 \cdot e^{\frac{-i2\pi nt}{T}}dt
\frac{1}{T}\left[\frac{T e^{\frac{-i2\pi nt}{T}}}{i2\pi n}\right]^{\frac{T}{2}}_0
\frac{1}{T}\left[\left(\frac{T e^{-in\pi}}{in2\pi}\right)-\left(\frac{T}{in2\pi}\right)\right]
\left(\frac{e^{-in\pi}}{in2\pi}-\frac{1}{in2\pi}\right)
However, the expression the author finds is:
\frac{1}{i\pi n} for odd n, and zero for even n.
First, I understand that the coefficient c_n=\frac{1}{T}\int^T_0 f(t) e^{\frac{-i2\pi nt}{T}}dt
Given a square wave, of period 2T, and amplitude :
http://s90.photobucket.com/user/jonnburton/media/SqareWave_zpsbdf193c1.jpg.html
It can be seen directly from the graph that c_0=\frac{1}{2} (I also managed to calculate this, too.)
But I can't follow how the author obtained expressions for c_n.
This is what I have done in attempt to obtain c_n:
c_n=\frac{1}{T}\int^{\frac{T}{2}}_0 1 \cdot e^{\frac{-i2\pi nt}{T}}dt
\frac{1}{T}\left[\frac{T e^{\frac{-i2\pi nt}{T}}}{i2\pi n}\right]^{\frac{T}{2}}_0
\frac{1}{T}\left[\left(\frac{T e^{-in\pi}}{in2\pi}\right)-\left(\frac{T}{in2\pi}\right)\right]
\left(\frac{e^{-in\pi}}{in2\pi}-\frac{1}{in2\pi}\right)
However, the expression the author finds is:
\frac{1}{i\pi n} for odd n, and zero for even n.
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