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 = x_{0} is a rational expression
approximating f. The parameters p and a_{0} are given by the leading order term f = a_{0} (x  x_{0})^{p} + O((x  x_{0})^{p + 1}) of the series expansion of f around x = x_{0}. The parameters a_{1}, …, b_{n} 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  x_{0} must
exist.
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:
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])
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 

An identifier 

An arithmetical
expression. If 

A list of nonnegative integers specifying the order of the approximation.
The default values are 
Arithmetical expression or FAIL
.