principal part#Calculus

The principal part at $z=a$ of a function

: $f(z)\; =\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; a\_k\; (z-a)^k$

is the portion of the Laurent series consisting of terms with negative degree.{{cite book | url=https://books.google.com/books?id=_cADk52kr4oC&dq=%22is+the+portion+of+the+Laurent+series+consisting+of+terms+with+negative+degree.%22&pg=PT48 | title=Laurent | date=16 October 2016 | isbn=9781467210782 | accessdate=31 March 2016}} That is,

: $\backslash sum\_\{k=1\}^\backslash infty\; a\_\{-k\}\; (z-a)^\{-k\}$

is the principal part of $f$ at $a$.

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

Consider the difference between the function differential and the actual increment:

:$\backslash frac\{\backslash Delta\; y\}\{\backslash Delta\; x\}=f\text{'}(x)+\backslash varepsilon$

:$\backslash Delta\; y=f\text{'}(x)\backslash Delta\; x\; +\backslash varepsilon\; \backslash Delta\; x\; =\; dy+\backslash varepsilon\; \backslash Delta\; x$

The differential dy is sometimes called the principal (linear) part of the function increment Δy.

The term principal part is also used for certain kinds of distributions having a singular support at a single point.

• Mittag-Leffler's theorem
• Cauchy principal value

{{Reflist}}

• [http://planetmath.org/encyclopedia/CauchyPrinciplePartIntegral.html Cauchy Principal Part at PlanetMath]

Category:Complex analysis

Category:Generalized functions