A Kapteyn series is a series of the form sum_(n = 0)^∞ α_n J_(ν + n)[(ν + n) z], where J_n(z) is a Bessel function of the first kind. Examples include Kapteyn's original series 1/(1 - z) = 1 + 2 sum_(n = 1)^∞ J_n(n z) and z^2/(2(1 - z^2)) = sum_(n = 1)^∞ J_(2n)(2n z).

Bessel function Neumann series | Bessel function of the first kind | Fourier-Bessel series | generalized Fourier series | lemon surface | Schlömilch's series