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.
