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The fourier series can (among others) expressed in terms of sines and cosines with coefficients [itex]a_n[/itex] and [itex]b_n[/itex] and solely by sines using amplitudes [itex]A_n[/itex] and phase [itex]\phi_n[/itex].

I want to express the latter using [itex]a_n[/itex] and [itex]b_n[/itex]. Using

[tex]

a_n = A_n \sin(\phi_n) \\

b_n = A_n \cos(\phi_n)

[/tex]

I quickly found [itex]A_n[/itex] by expressing the arccos and arcsin. For [itex]\phi_n[/itex] I would get

[tex]\phi_n = \arccos \frac{a_n}{\sqrt{a_n^2 + b_n^2}}[/tex]

However, according to the German Wikipedia (http://de.wikipedia.org/wiki/Fourier-Reihe) this seems not so trivial. One option there is (for n \neq 0):

[tex]\phi_n = 2 \arctan \frac{b_n}{A_n + a_n}[/tex]

or using arctan and signum function. What I am missing or is my approach also correct?

Thanks,

divB

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# Fourier series: relation of coefficients

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