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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
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Properties & Relations  
Possible Issues  
See Also
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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 a discrete weighted measure:

Plot the first few polynomials:

Verify the orthogonality of the polynomials with respect to the 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:

Wolfram Research (2007), PadeApproximant, Wolfram Language function, https://reference.wolfram.com/language/ref/PadeApproximant.html.

Text

Wolfram Research (2007), PadeApproximant, Wolfram Language function, https://reference.wolfram.com/language/ref/PadeApproximant.html.

CMS

Wolfram Language. 2007. "PadeApproximant." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PadeApproximant.html.

APA

Wolfram Language. (2007). PadeApproximant. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PadeApproximant.html

BibTeX

@misc{reference.wolfram_2025_padeapproximant, author="Wolfram Research", title="{PadeApproximant}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/PadeApproximant.html}", note=[Accessed: 12-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_padeapproximant, organization={Wolfram Research}, title={PadeApproximant}, year={2007}, url={https://reference.wolfram.com/language/ref/PadeApproximant.html}, note=[Accessed: 12-August-2025]}