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.

# 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 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: