convert/FormalPowerSeries  convert to formal power (or LaurentPuiseux) series

Calling Sequence


convert(expr, FormalPowerSeries, eq, k, opts)
convert(expr, FormalPowerSeries, eq, b(k), opts)


Parameters


expr



algebraic expression

eq



equation (e.g. x=a) or name (e.g. x); optional if expr contains only one variable

k



(optional) name of the summation variable in the result

b(k)



(optional) name for the kth series coefficient

opts



sequence of options of the form keyword=value; possible keywords are method, makereal, dir, differentialorder, and recurrence





Options


•

differentialorder: a positive integer n (default: n=4); upper bound for the order of the differential equation searched for. This controls the depth of the search for a differential equation for expr. Higher values of n will increase the chance to find the solution, but increase the running time as well.

•

dir: one of default, left, right, real, or complex; direction of the limit computation for initial values. If a is finite, then the default is dir=complex. If a is either infinity or infinity, then the default is dir=real. See also limit.

•

makereal: either true or false (default); makereal=true (or just makereal for short) indicates that a series with real coefficients should be returned

•

method: one of default, hypergeometric, rational, or exponential. Specifies the method that will be used; the default method uses an internal selection strategy. See the Examples below for an illustration of the various methods.

•

recurrence: either true or false (default). If recurrence=true (or recurrence for short) is given and no formal power series can be computed, then the output is a recurrence for .



Description


•

This command expands meromorphic functions of certain type into their corresponding LaurentPuiseux series as a sum of terms of the form , where m is called the symmetry number, s is the shift number, and a is the expansion point.

•

The following types are supported:

–

functions of hypergeometric type, where is a rational function of k for some integer m;

–

functions of exponential type, which satisfy a linear homogeneous differential equation with constant coefficients;

–

functions of rational type, which are either rational or have a rational derivative;

–

linear combinations of hypergeometric functions are treated by the PetkovsekvanHoeij algorithm; see LREtools[hypergeomsols].

•

The convert(expr, FormalPowerSeries, x=a) command tries to find a formal power series expansion for expr with respect to the variable x at the point of expansion a. If a=infinity, then the command searches for an asymptotic series. It also works for formal LaurentPuiseux series, and in certain cases of logarithmic singularities.

•

The command first looks for a homogeneous linear differential equation with polynomial coefficients for expr; hence Maple must know the derivatives of expr.

•

If eq is a variable name x, or if eq is omitted and expr has only one variable x, then x=0 is assumed.

•

The optional argument k is a name that will be taken as the summation variable in the result. If it is not specified, then one of the variable names k, k0, k1, etc. is chosen.

•

To compute asymptotic power series, one may expand the function around ; see the Examples below. The result is a (possibly divergent) series.

•

The FormalPowerSeries argument can be abbreviated as FPS.

•

For a complete list of known functions, see inifcns.

•

The convert/sum command provides the same functionality as the convert/FormalPowerSeries command, with a newer algorithm which can employ alternate methods.



Examples


>


 (1) 
>


 (2) 
>


 (3) 
>


 (4) 
>


 (5) 
The following examples illustrate the use of the method parameter.
>


 (6) 
>


 (7) 
>


 (8) 
>


 (9) 
The output can be a Puiseux series or a Laurent series.
>


 (10) 
>


 (11) 
Userdefined functions are handled provided their derivative is known. We define the derivative of the function as follows (see diff for more information).
>

`diff/g` := proc(a,x) g(a)*diff(a,x) end proc:

>


 (12) 
Indefinite integrals are handled.
>


 (13) 
Linear combinations of hypergeometric functions are recognized.
>


 (14) 
The input functions can contain parameters.
>


 (15) 
In the next example, the output is expressed in terms of algebraic numbers.
>


 (16) 
Maple's special functions are handled.
>


 (17) 
>


 (18) 
Hidden polynomials are detected.
>


 (19) 
>


 (20) 
>


 (21) 
Asymptotic power series can be computed.
>


 (22) 
>


 (23) 
Real and onesided (asymptotic) series can be computed using the dir option.
>


 (24) 
>


 (25) 
>


 (26) 
Some examples where convert(...,FormalPowerSeries) does not succeed, e.g., because of an essential singularity.
>


 (27) 
>


 (28) 
Here is an example where convert(...,FormalPowerSeries) fails to compute a formal power series, but is able to determine a recurrence equation for the coefficients.
>


 (29) 
>


 (30) 
Generalized series.
>


 (31) 


See Also


convert/0F1, convert/1F1, convert/2F1, convert/hypergeom, convert/sum, convert/to_special_function, gfun[holexprtodiffeq], hypergeom, LREtools[hypergeomsols], series, simplify/hypergeom, sum, SumTools, taylor


References



Gruntz, Dominik, and Koepf, Wolfram. "Maple package of formal power series." Maple Technical Newletter, Vol. 2(2), (1995):2228.


Koepf, Wolfram. "Algorithmic development of power series. Artificial intelligence and symbolic mathematical computing." Lecture Notes in Computer Science, Vol. 737, pp. 195213. Edited by J. Calmet and J. A. Campbell. BerlinHeidelberg: Springer, 1993.


Koepf, Wolfram. "Examples for the algorithmic calculation of formal Puiseux, Laurent and power series." SIGSAM Bulletin Vol. 27, (1993): 2032.


Koepf, Wolfram. "Power series, Bieberbach conjecture and the de Branges and Weinstein functions." Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation. pp. 169175. New York: ACM, 2003.


Koepf, Wolfram. "Power series in computer algebra." Journal of Symbolic Computation, Vol. 13, (1992): 581603.


van Hoeij, Mark. "Finite singularities and hypergeometric solutions of linear recurrence equations." J. Pure and Appl. Algebra, Vol. 139, (1999): 109131.


