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NSeries


gives a numerical approximation to the series expansion of f about the point including the terms through .
  • 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.
  • 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:
Radius1radius of circle on which f is sampled
WorkingPrecisionMachinePrecisionprecision used in internal computations
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]=
Find expansions in the complex plane:
Find Laurent expansions about essential singularities:
Series will not find Laurent expansions about essential singularities:
Use Radius to pick the annulus within which the Laurent series will converge:
Laurent series for :
Changing Radius can improve accuracy:
A function defined only for numerical input:
Find a series expansion of f:
Check:
NResidue can also be used to construct a series of a numerical function:
Using NResidue:
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:
Find the series expansion of the generating function for unrestricted partitions:
Check: