Bessel series for a constant

Fourier series express functions as a sum of sines and cosines of different frequencies. Bessel series are analogous, expressing functions as a sum of Bessel functions of different orders.

Fourier series arise naturally when working in rectangular coordinates. Bessel series arise naturally when working in polar coordinates.

The Fourier series for a constant is trivial. You can think of a constant as a cosine with frequency zero.

The Bessel series for a constant is not as simple, but more interesting. Here we have

1 = J_0(x) + 2J_2(x) + 2J_4(x) + 2J_6(x) + \cdots

Since

J_{-n}(x) = (-1)^n J_n(x)

we can write the series above more symmetrically as

1 = \sum_{n=-\infty}^\infty J_{2n}(x)

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