  The general features and characteristics of Kapteyn series, which are a
special type of series involving Bessel function, are investigated. For many
applications to physics, astrophysics, and mathematics, it is crucial to have
closed-form expressions in order to determine their functional structure and
parametric behavior. Closed-form expressions of Kapteyn series have mostly been
limited to special cases, even though there are often similarities in the
approaches used to reduce the series to analytically tractable forms. The goal
of this paper is to review the previous work in the area and to show that
Kapteyn series can be expressed as trigonometric or gamma function series,
which can be evaluated in closed form for specific parameters. Two examples
with a similar structure are given, showing the complexity of Kapteyn series.
