Principal part

From Companal

Definition

Suppose is an open subset and . Suppose is a holomorphic function, so is an isolated singularity of . The principal part of at is defined in the following equivalent ways:

  • It is the function given in a neighborhood of by the part of the Laurent series for about , comprising only the negative powers.
  • It is the unique function (upto germ equivalence at ) such that is a holomorphic function, and such that has no nonnegative powers in its Laurent series expansion.