Built-in Mathematica Symbol

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.

Mathematica 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.

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

PadeApproximant can handle functions with poles:

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

PadeApproximant can handle functions with poles:

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:

Padé approximant centered at the point x=a:

Padé approximant in fractional powers:

Padé approximant of a function containing logarithmic terms:

Plot successive Padé approximants to

:

Construct discrete orthogonal polynomials with respect to discrete weighted measure:

The Padé approximant agrees with the ordinary series for

terms:

For

PadeApproximant gives an ordinary series:

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: