A Fourier series -like expansion of a twice continuously
differentiable function
(1)
for Bessel function of
the first kind . The coefficients are then given by
(Gradshteyn and Ryzhik 2000, p. 926), where
As an example, consider
so
(9)
(Whittaker and Watson 1990, p. 378; Gradshteyn and Ryzhik 2000, p. 926). This is illustrated above with 1 (red), 2 (green), 3 (blue), and 4 terms (violet) included.
Similarly, for
(10)
See also Bessel Function of the First Kind ,
Fourier-Bessel Series ,
Fourier Series
Explore with Wolfram|Alpha
References Chapman. Quart. J. 43 , 34-37, 1912. Gradshteyn, I. S. and Ryzhik, I. M. "The Series Tables
of Integrals, Series, and Products, 6th ed. Itô, K. (Ed.). Encyclopedic
Dictionary of Mathematics, 2nd ed., Vol. 2. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic
Dictionary of Mathematics. Schlömilch.
Z. für Math. Phys. 3 , 137-165, 1857. Whittaker, E. T.
and Watson, G. N. "Schlömilch's Expansion of an Arbitrary Function
in a Series of Bessel Coefficients of Order Zero." §17.82 in A
Course in Modern Analysis, 4th ed. Referenced on Wolfram|Alpha Schlömilch's Series
Cite this as:
Weisstein, Eric W. "Schlömilch's Series."
From MathWorld https://mathworld.wolfram.com/SchloemilchsSeries.html
Subject classifications
,
where 
is a zeroth order Bessel function of
the first kind. The coefficients are then given by
and care should be taken to avoid the two typos
of Iyanaga and Kawada (1980) and Itô (1986).
, which has 
and therefore
,
."