Fourier series expansion error?

[Anton Makarov]

Hi, I'm trying to find fourier series decomposition of 1/x function on [-3; 3] interval. IMHO, such a decomposition doesn't exist, because the a0 и an doesn't exist (corresponding integrals doesn't converge). But sympy function fourier_series, returns the answer (omitting the a0, an coefficients and only taking into account the bn coefficients). Is there an error in fourier_series function?

fourier_series(1/x,(x,-3,3)) = (2/3)*sin(pi*x/3)*Si(pi)+(2/3)*sin(2*pi*x/3)*Si(2*pi)+(2/3)*sin(pi*x)*Si(3*pi)+...

[emanuel.c...@gmail.com]

Let’s dissecate this :

>>> from sympy import symbols, fourier_series
>>> t=symbols("t", real=True)
>>> T=symbols("T", positive=True)
>>> foo=fourier_series(1/t, (t, -T/2, T/2)) ; foo
FourierSeries(1/t, (t, -T/2, T/2), (0, SeqFormula(0, (_k, 1, oo)), SeqFormula(4*sin(2*_n*pi*t/T)*Si(_n*pi)/T, (_n, 1, oo))))

The constant term tof this series is taken to be 0. The even terms are all 0. The coefficient of the nth odd term is Si(n*pi). These coefficients do not converge to 0 :

>>> k=symbols("k", integer=True)
>>> limit(Si(k*pi), k, oo)
pi/2

therefore their sum does not necessarily converge.

Therefore, the expressions returned by fourier_transform are analitically correct, but their existence do not imply their convergence.

In other word, fourier series return the elements allowing computation of the Fourier series of the function if such a series exist, but do not assess its existence.

HTH,

​

[Антон Макаров]

I understand, thank you for suck a deep explanation…

Отправлено из мобильной Почты Mail.ru

суббота, 17 декабря 2022 г., 17:33 +0300 от emanuel.c...@gmail.com <emanuel.c...@gmail.com>:

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