NUMERICAL CALCULUS PACKAGE SYMBOL

NSeries

gives a numerical approximation to the series expansion of f about the point

including the terms

through

.

MORE INFORMATION

To use , you first need to load the Numerical Calculus Package using .

The function f must be numeric when its argument x is numeric.

will construct standard univariate Taylor or Laurent series.

samples f at points on a circle in the complex plane centered at 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.

will not return a correct result if the disk centered at contains a branch cut of f.

The result of is a SeriesData object.

If the result of is a Laurent series, than 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.

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 x=0:

Chop is needed to eliminate spurious residuals:

Using extended precision may also eliminate spurious imaginaries:

Needs["NumericalCalculus`"]

This is a power series for the exponential function around x=0:

In[2]:=

Out[2]=

Chop is needed to eliminate spurious residuals:

In[3]:=

Out[3]=

Using extended precision may also eliminate spurious imaginaries:

In[4]:=

Out[4]=

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

:

Changing Radius can improve accuracy:

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)

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

:

Here is the SeriesData representation of the Laurent series of Exp[

+x]²:

Neat Examples (1)

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

Check: