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This package can expand a specific class of functions into their corresponding Laurent-Puiseux series.3
This package can expand functions of certain type into their corresponding Laurent-Puiseux series as a sum of terms of the form
textbffunctions of ‘rational type’, which are either rational or have a rational derivative of some order;
functions of ‘hypergeometric type’ where \(a(k+m)/a(k)\) is a rational function for some integer \(m\);
functions of ‘explike type’ which satisfy a linear homogeneous differential equation with constant coefficients.
The FPS package is an implementation of the method presented in [Koe92]. The implementations of this package for MAPLE (by D. Gruntz) and MATHEMATICA (by W. Koepf) served as guidelines for this one.
Numerous examples can be found in [Koe93b, Koe93a], most of which are contained in
the test file fps.tst. Many more examples can be found in the extensive bibliography
of Hansen [Han75].
FPS fps(f,x,x0) tries to find a formal power series expansion for f with respect to the
variable x at the point of development x0. It also works for formal Laurent (negative
exponents) and Puiseux series (fractional exponents). If the third argument is omitted,
then x0:=0 is assumed.
Examples: fps(asin(x)^2,x) results in
2*k 2*k 2 2 x *2 *factorial(k) *x infsum(----------------------------,k,0,infinity) factorial(2*k + 1)*(k + 1)
fps(sin x,x,pi) gives
2*k k ( - pi + x) *( - 1) *( - pi + x) infsum(------------------------------------,k,0, factorial(2*k + 1) infinity)
and fps(sqrt(2-x^2),x) yields
2*k - x *sqrt(2)*factorial(2*k) infsum(--------------------------------,k,0,infinity) k 2 8 *factorial(k) *(2*k - 1)
Note: The result contains one or more infsum terms such that it does not interfere
with the REDUCE operator sum. In graphical oriented REDUCE interfaces this operator
results in the usual \(\sum \) notation.
If possible, the output is given using factorials. In some cases, the use of the
Pochhammer symbol pochhammer(a,k)\(:=a(a+1)\cdots (a+k-1)\) is necessary.
The operator fps uses the operator SimpleDE of the next section.
If an error message of type
Could not find the limit of:
occurs, you can set the corresponding limit yourself and try a recalculation. In the
computation of fps(atan(cot(x)),x,0), REDUCE is not able to find the value
for the limit limit(atan(cot(x)),x,0) since the atan function is multi-valued.
One can choose the branch of atan such that this limit equals \(\pi /2\) so that we may
set
let limit(atan(cot(~x)),x,0)=>pi/2;
and a recalculation of fps(atan(cot(x)),x,0) yields the output pi - 2*x
which is the correct local series representation.
SimpleDESimpleDE(f,x) tries to find a homogeneous linear differential equation
with polynomial coefficients for \(f\) with respect to \(x\). Make sure that \(y\) is not a used
variable. The setting factor df; is recommended to receive a nicer output
form.
Examples: SimpleDE(asin(x)^2,x) then results in
2 df(y,x,3)*(x - 1) + 3*df(y,x,2)*x + df(y,x)
SimpleDE(exp(x^(1/3)),x) gives
2 27*df(y,x,3)*x + 54*df(y,x,2)*x + 6*df(y,x) - y
and SimpleDE(sqrt(2-x^2),x) yields
2 df(y,x)*(x - 2) - x*y
The depth for the search of a differential equation for f is controlled by the variable
fps_search_depth; A higher value for fps_search_depth will increase the
chance to find the solution, but increases the complexity as well. The default
value for fps_search_depth is \(5\). E. g., for fps(sin(x^(1/3)),x), or
SimpleDE(sin(x^(1/3)),x) a setting fps_search_depth:=6 is
necessary.
The output of the FPS package can be influenced by the switch tracefps. Setting on
tracefps causes various prints of intermediate results.
The handling of logarithmic singularities is not yet implemented.
The rational type implementation is not yet complete.
The support of special functions [Koe94a] will be part of the next version.
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