NumericalCalculus`
NumericalCalculus`

# NSeries

NSeries[f,{x,x0,n}]

gives a numerical approximation to the series expansion of f about the point x=x0 including the terms (x-x0)-n through (x-x0)n.

# Details and Options

• To use NSeries, you first need to load the Numerical Calculus Package using Needs["NumericalCalculus`"].
• The function f must be numeric when its argument x is numeric.
• NSeries will construct standard univariate Taylor or Laurent series.
• NSeries samples f at points on a circle in the complex plane centered at x0 and uses InverseFourier. The option Radius specifies the radius of the circle.
• The region of convergence will be the annulus (containing the sampled points) where f is analytic.
• NSeries will not return a correct result if the disk centered at x0 contains a branch cut of f.
• The result of NSeries is a SeriesData object.
• If the result of NSeries is a Laurent series, then the SeriesData object is not a correct representation of the series, as higher-order poles are neglected.
• No effort is made to justify the precision in each of the coefficients of the series.
• NSeries is unable to recognize small numbers that should in fact be zero. Chop is often needed to eliminate these spurious residuals.
• The number of sample points chosen is .
• The following options can be given:
•  Radius 1 radius of circle on which f is sampled WorkingPrecision MachinePrecision precision used in internal computations

# Examples

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## Basic Examples(1)

This is a power series for the exponential function around :

Chop is needed to eliminate spurious residuals:

Using extended precision may also eliminate spurious imaginaries:

## Scope(2)

Find expansions in the complex plane:

Find Laurent expansions about essential singularities:

Series will not find Laurent expansions about essential singularities:

## Options(2)

Use Radius to pick the annulus within which the Laurent series will converge:

Laurent series for :

## Applications(1)

A function defined only for numerical input:

Find a series expansion of f:

Check:

## Properties & Relations(1)

NResidue can also be used to construct a series of a numerical function:

Using NResidue:

## Possible Issues(2)

NSeries can have aliasing problems due to InverseFourier:

The correct expansion is analytic at the origin:

SeriesData cannot correctly represent a Laurent series. Here is the square of the series of Exp[ +x]:

Here is the SeriesData representation of the Laurent series of :

## Neat Examples(1)

Find the series expansion of the generating function for unrestricted partitions:

Check: