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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.
  • 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.
  • If the result of NSeries 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.
  • 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 22+LeftCeilinglog2[n]RightCeiling.
  • The following options can be given:
Radius1radius of circle on which f is sampled
WorkingPrecisionMachinePrecisionprecision used in internal computations
EXAMPLESCLOSE ALL
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]:=
Click for copyable input
Out[2]=
Chop is needed to eliminate spurious residuals:
In[3]:=
Click for copyable input
Out[3]=
Using extended precision may also eliminate spurious imaginaries:
In[4]:=
Click for copyable input
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 x≥3:
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)
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 :
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:
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