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

Laurent series definition

The principal part at

z=a

of a function :

f(z) = \sum_^\infty a_k (z-a)^k

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

\sum_^\infty a_ (z-a)^

is the principal part of

f

a

. If the Laurent series has an inner radius of convergence of 0 , then

f(z)

has an

essential singularity In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...

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.

Other definitions

Calculus

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

\frac=f'(x)+\varepsilon

\Delta y=f'(x)\Delta x +\varepsilon \Delta x = dy+\varepsilon \Delta x

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

Distribution theory

The term principal part is also used for certain kinds of distributions having a

singular support In mathematics, the support of a Real number, real-valued Function (mathematics), function f is the subset of the function Domain of a function, domain containing the elements which are not mapped to zero. If the domain of f is a topological spac ...

at a single point.

References

{{Reflist

External links