Principal part

In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.

The principal part at ${\displaystyle z=a}$ of a function

is the portion of the Laurent series consisting of terms with negative degree. That is,

is the principal part of ${\displaystyle f}$ at ${\displaystyle a}$. ${\displaystyle f(z)}$ has an essential singularity at ${\displaystyle a}$, if and only if the principal part is an infinite sum.

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