PadeApproximant

PadeApproximant[expr,{x,x0,{m,n}}]

gives the Padé approximant to expr about the point x=x0, with numerator order m and denominator order n.

PadeApproximant[expr,{x,x0,n}]

gives the diagonal Padé approximant to expr about the point x=x0 of order n.

Details

  • The Wolfram Language can find the Padé approximant about the point x=x0 only when it can evaluate power series at that point.
  • PadeApproximant produces a ratio of ordinary polynomial expressions, not a special SeriesData object.

Examples

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Basic Examples  (2)

Order [2/3] Padé approximant for Exp[x]:

PadeApproximant can handle functions with poles:

Scope  (4)

Padé approximant of an arbitrary function:

Padé approximant with a complex-valued expansion point:

Padé approximant with an expansion point at infinity:

Find a Padé approximant to a given series:

Generalizations & Extensions  (3)

Padé approximant centered at the point :

Padé approximant in fractional powers:

Padé approximant of a function containing logarithmic terms:

Applications  (2)

Plot successive Padé approximants to :

Construct discrete orthogonal polynomials with respect to discrete weighted measure:

Properties & Relations  (2)

The Padé approximant agrees with the ordinary series for terms:

For PadeApproximant gives an ordinary series:

Possible Issues  (2)

Padé approximants often have spurious poles not present in the original function:

Padé approximants of a given order may not exist:

Perturbing the order slightly is usually sufficient to produce an approximant:

Introduced in 2007
 (6.0)