We know that because [itex] \sin{nx} [/itex] and [itex] \cos{nx} [/itex] are degenerate eigenfunctions of a hermition operator(the SHO equation),and eachof them form a complete set so we for every [itex] f(x) [/itex] ,we have:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

f(x)=\frac{a_0}{2}+\Sigma_1^{\infty} a_n \cos{nx}

[/itex]

and

[itex]

f(x)=\Sigma_1^{\infty} b_n \sin{nx}

[/itex]

But here,because for every [itex] m [/itex],[itex] \sin{mx} [/itex] and [itex] \cos{mx} [/itex] are orthogonal,we also can have:

[itex]

f(x)=\frac{a_0}{2}+\Sigma_1^{\infty} a_n \cos{nx} + \Sigma_1^{\infty} b_n \sin{nx}

[/itex]

And its easy to understand that the [itex] a_n [/itex]s and [itex] b_n [/itex]s are the same.

So it seems we reach to a paradox!

What's wrong?

thanks

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# A question about Fourier series

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