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


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


  • 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.


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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,


Wolfram Research (2007), PadeApproximant, Wolfram Language function,


Wolfram Language. 2007. "PadeApproximant." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). PadeApproximant. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_padeapproximant, author="Wolfram Research", title="{PadeApproximant}", year="2007", howpublished="\url{}", note=[Accessed: 24-June-2024 ]}


@online{reference.wolfram_2024_padeapproximant, organization={Wolfram Research}, title={PadeApproximant}, year={2007}, url={}, note=[Accessed: 24-June-2024 ]}