Principal part (of a Laurent series)

The negative-power half of a Laurent series, $\sum_{n=1}^\infty a_{-n}(z-z_0)^{-n}$ .

The principal part of a Laurent series $\sum_{n=-\infty}^\infty a_n(z-z_0)^n$ is the sum of its negative-power terms, $\sum_{n=1}^\infty a_{-n}(z-z_0)^{-n}$ ; the remaining non-negative-power terms form its analytic part. The principal part carries all the information about a singularity at $z_0$ : whether it is removable, a pole, or essential is read directly off how many (if any) of its terms are nonzero.

Examples

The Laurent series $\frac1{z^2}+\frac1z+1+z$ has principal part $\frac1{z^2}+\frac1z$ and analytic part $1+z$ .
A function's principal part at $z_0$ is identically zero exactly when the singularity at $z_0$ is removable.