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Pade approximation

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pade(f, x, <[m, n]>)
pade(f, x = x0, <[m, n]>)


pade(f, ...) computes a Pade approximant of the expression f.

The Pade approximant of order [m, n] around x = x0 is a rational expression

approximating f. The parameters p and a0 are given by the leading order term f = a0 (x - x0)p + O((x - x0)p + 1) of the series expansion of f around x = x0. The parameters a1, …, bn are chosen such that the series expansion of the Pade approximant coincides with the series expansion of f to the maximal possible order.

The expansion points infinity, -infinity, and complexInfinity are not allowed.

If no series expansion of f can be computed, then FAIL is returned. Note that series must be able to produce a Taylor series or a Laurent series of f, i.e., an expansion in terms of integer powers of x - x0 must exist.


Example 1

The Pade approximant is a rational approximation of a series expansion:

f := cos(x)/(1 + x): P := pade(f, x, [2, 2])

For most expressions of leading order 0, the series expansion of the Pade approximant coincides with the series expansion of the expression through order m + n:

S := series(f, x, 6)

This differs from the expansion of the Pade approximant at order 5:

series(P, x, 6)

The series expansion can be used directly as input to pade:

pade(S, x, [2, 3]), pade(S, x, [3, 2])

Both Pade approximants approximate f through order m + n = 5:

map([%], series, x)

delete f, P, S:

Example 2

The following expression does not have a Laurent expansion around x = 0:

series(x^(1/3)/(1 - x), x)

Consequently, pade fails:

pade(x^(1/3)/(1 - x), x, [3, 2])

Example 3

Note that the specified orders [m, n] do not necessarily coincide with the orders of the numerator and the denominator if the series expansion does not start with a constant term:

pade(x^10*exp(x), x, [2, 2]), pade(x^(-10)*exp(x), x, [2, 2])



An arithmetical expression or a series of domain type Series::Puiseux generated by the function series


An identifier


An arithmetical expression. If x0 is not specified, then x0 = 0 is assumed.

[m, n]

A list of nonnegative integers specifying the order of the approximation. The default values are [3, 3].

Return Values

Arithmetical expression or FAIL.

See Also

MuPAD Functions

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