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.
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
open all close allBasic 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:
Tech Notes
Related Guides
History
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: 16-November-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: 16-November-2025]}