Let any finite or infinite set of points having no finite limit point be prescribed and associate with each of its points a principal part, i.e.,
a rational function of the special form

for , 2, ..., . Then there exists a meromorphic
function which has poles with the prescribed principal parts at precisely the
prescribed points, and is otherwise regular. It can be represented in the form of
a partial fraction decomposition from which one can read off again the poles, along
with their principal parts. Further, if is one such function, then

is the most general function satisfying the conditions of the problem, where
denotes an arbitrary entire function.