Laurent series

The formal Laurent series over the complex numbers is 'not' a local field.

This can also lead to interesting identities if a Laurent series is already known.

This is a Laurent series with finite principal part.

This completion can be described as the field of formal Laurent series over .

Under certain conditions the complete expansion as a Laurent series can be obtained:

It is the algebraic closure of the field of Laurent series.

This function has the following Laurent series about the essential singular point at 0:

Z-transform - the special case where the Laurent series is taken about zero has much use in time series analysis.

Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities.

The Laurent series for a complex function f(z) about a point c is given by:

Laurent series cannot in general be multiplied.

An example from category 1 above is the field of Laurent series with a finite number of negative-power terms.

Geometrically, the two Laurent series may have non-overlapping annuli of convergence.

Alternatively, residues can be calculated by finding Laurent series expansions, and are sometimes defined in terms of them.

One may define formal differentiation for formal Laurent series in a natural way (term-by-term).

Hence the Laurent series is unique.

The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.

Laurent series: An extension of the Taylor series, allowing negative exponent values.

Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.

Outside the annulus, the Laurent series diverges.

In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers.

Extending a result from classical function theory, R has a Laurent series representation on A:

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 Levi-Civita field is similar to the Laurent series, but is algebraically closed.

Examples of function series include power series, Laurent series, Fourier series, etc.