Inmathematics, theprincipal part has several independent meanings but usually refers to the negative-power portion of theLaurent series of a function.

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

${\displaystyle f(z)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_k(z-a{)}^k}$

is the portion of theLaurent series consisting of terms with negative degree.^[1] That is,

${\displaystyle \sum _{k=1}^{\mathrm{\infty }}{a}_{-k}(z-a{)}^{-k}}$

is the principal part of${\displaystyle f}$ at${\displaystyle a}$.If the Laurent series has an inner radius of convergence of${\displaystyle 0}$, then${\displaystyle f(z)}$ has anessential singularity at${\displaystyle a}$ if and only if the principal part is an infinite sum. If the inner radius of convergence is not${\displaystyle 0}$, then${\displaystyle f(z)}$ may be regular at${\displaystyle a}$ despite the Laurent series having an infinite principal part.

Consider the difference between the functiondifferential and the actual increment:

${\displaystyle \frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=f^{\prime }(x)+\epsilon }$

${\displaystyle \mathrm{\Delta }y=f^{\prime }(x)\mathrm{\Delta }x+\epsilon \mathrm{\Delta }x=dy+\epsilon \mathrm{\Delta }x}$

The differentialdy is sometimes called theprincipal (linear) part of the function incrementΔy.

The termprincipal part is also used for certain kinds ofdistributions having asingular support at a single point.

• Mittag-Leffler's theorem
• Cauchy principal value

• Cauchy Principal Part atPlanetMath.

• Complex analysis
• Generalized functions

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