why are there 2 exponential terms on right side of equation in red box ?
i know euler identity but why does ,the left side is trigonometry form when n=1 equal to exponential form n=1 + exponential form n=-1
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\$\begingroup\$ Do you know that, for example, \$e^{j\omega t}=cos(\omega t) +j\:sin(\omega t)\$? \$\endgroup\$– ChuCommented Jul 13, 2019 at 23:53
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\$\begingroup\$ i know this identity. \$\endgroup\$– Jittinan SuwanrueangsriCommented Jul 14, 2019 at 2:23
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\$\begingroup\$ ...so, the \$\small C_{-1}\$ and \$\small C_1\$ coefficients are complex conjugates, and when multiplied by the complex exponentials in the manner indicated in the red box, give the alternative trigonometric form of the harmonic Fourier series. \$\endgroup\$– ChuCommented Jul 14, 2019 at 8:32
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2 Answers
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The n= 1 trig form must equal the sum of the +/-1 exponential terms because those two exponentials are the only terms in that entire infinite summation with the same period (frequency). Without even thinking about the math and the fact that sinusoids of different frequencies are orthogonal, at a purely intuitive level, if two frequencies are to be equal, they must have the same period and for a given n in the trig series, only the +/- terms in the exponential series have the same period. Therefore that expression must be true.
If you want to understand at a deeper level what that means, you'll get to that once you learn about the Fourier relationship between even/odd functions and positive/negative frequencies.
answered Jul 14, 2019 at 2:38
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This equation contains ALL fundamental frequency terms. It's written to find how a's and b's are connected to c's.
In the red box the left side is a general formula how to make an arbitary real valued sinusoidal signal with arbitary phase angle by summing sine and cosine which both have ZERO phase angle.
In the right side there is a sum of complex numbers to get rid of the imaginary part. This is the start of the derivation the relation, so it's not said that the right side must have total imaginary part=0, but it will surely be said later.
Hopefully you know Euler's identity. Without it this stays as a piece of mumbojumbo.