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convert/FormalPowerSeries

convert to formal power (or Laurent-Puiseux) series

 

Calling Sequence

Parameters

Options

Description

Examples

References

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 bk.

Description

• 

This command expands meromorphic functions of certain type into their corresponding Laurent-Puiseux series as a sum of terms of the form k=0bkxakm+sq, 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 bk+mbk 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 Petkovsek-van-Hoeij 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 Laurent-Puiseux 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

convertsinx,FormalPowerSeries

k=0∞1kx2k+12k+1!

(1)

convertlnx,FormalPowerSeries,x=1,j

j=0∞1jx1j+1j+1

(2)

convert11xx,FormalPowerSeries,x,n

n=0∞1224n!16nxn2n!22n+1

(3)

convertⅇarcsinx,FormalPowerSeries

k=0∞j=0k4j2+1x2k4k2+12k!+k=0∞j=0k2j2+2j+12kx2k+12k2+2k+12k+1!

(4)

convertt1xtt2,FormalPowerSeries,t

k=0∞12x12x2+4kx2+4+12x+12x2+4kx2+4tk

(5)

The following examples illustrate the use of the method parameter.

convertln1+x21x,FormalPowerSeries,method=rational

k=0∞IkIkIIk+IIkxk+1IkIkk+1

(6)

convertⅇxsinx,FormalPowerSeries,method=hypergeometric

k=0∞12I1+Ikk!+12I1Ikk!xk

(7)

convertⅇx2ⅇx2cos3x2+π3,FormalPowerSeries,method=exponential

k=0∞cos23kπk!+3sin23kπk!+1k!xk

(8)

convertⅇx2ⅇx2cos3x2+π3,FormalPowerSeries,method=hypergeometric

k=0∞3x3k+13k+1!

(9)

The output can be a Puiseux series or a Laurent series.

convertⅇx,FormalPowerSeries

k=0∞xkk!

(10)

convertsinxx6,FormalPowerSeries

k=0∞1kx2k52k+1!

(11)

User-defined functions are handled provided their derivative is known. We define the derivative of the function g as follows (see diff for more information).

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

convertgx,FormalPowerSeries,x=0

k=0∞g0xkk!

(12)

Indefinite integrals are handled.

convert∫0xerfttⅆt,FormalPowerSeries,x

k=0∞21kx2k+1πk!2k+12

(13)

Linear combinations of hypergeometric functions are recognized.

convert1x2+1x36arctanx+1x24+112lnx2+1+5x212+14,FormalPowerSeries,x=0,makereal

k=0∞xk+4cos12kπk+4k+3k+2k+1

(14)

The input functions can contain parameters.

convertsinx+y,FormalPowerSeries,x

k=0∞siny1kx2k2k!+k=0∞cosy1kx2k+12k+1!

(15)

In the next example, the output is expressed in terms of algebraic numbers.

convert1x4+x+1,FormalPowerSeries,method=rational

k=0∞_α1=RootOf_Z4+_Z+11229xk36_α1348_α12+64_α1+27_α1k+1

(16)

Maple's special functions are handled.

convertAiryAix,FormalPowerSeries

k=0∞1331/39kx3kΓ23pochhammer23,kk!+k=0∞1231/6Γ239kx3k+1πpochhammer43,kk!

(17)

converthypergeoma,b,c,x,FormalPowerSeries,x

k=0∞pochhammerb,kpochhammera,kxkpochhammerc,kk!

(18)

Hidden polynomials are detected.

convertcos4arccosx,FormalPowerSeries

8x48x2+1

(19)

fexpandcosx2+sinx210

f:=cosx20+10cosx18sinx2+45cosx16sinx4+120cosx14sinx6+210cosx12sinx8+252cosx10sinx10+210cosx8sinx12+120cosx6sinx14+45cosx4sinx16+10cosx2sinx18+sinx20

(20)

convertf,FormalPowerSeries

1

(21)

Asymptotic power series can be computed.

convertⅇx21erfx,FormalPowerSeries,x=∞,dir=left

k=0∞1k2k!4k1x2k+1πk!

(22)

convertsin1x,FormalPowerSeries,x=∞

k=0∞1k1x2k+12k+1!

(23)

Real and one-sided (asymptotic) series can be computed using the dir option.

convertⅇ1x2,FormalPowerSeries,x=0,dir=real

0

(24)

convertⅇ1x,FormalPowerSeries,x=0,dir=right

0

(25)

convertⅇ1x,FormalPowerSeries,x=0,dir=left

ⅇ1x

(26)

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

convertⅇ1x,FormalPowerSeries

ⅇ1x

(27)

converttanx,FormalPowerSeries

tanx

(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.

convertarcsinx3,FormalPowerSeries

arcsinx3

(29)

convertarcsinx3,FormalPowerSeries,bk,recurrence

k4bk2k+1k+2k2+2k+2bk+2+k+1k+2k+3k+4bk+4=0

(30)

Generalized series.

convertarcsechx,FormalPowerSeries

ln2lnx+k=0∞122k+1!4kx2k+2k!2k+12k+2

(31)

References

  

Gruntz, Dominik, and Koepf, Wolfram. "Maple package of formal power series."  Maple Technical Newsletter, Vol. 2(2), (1995):22-28.

  

Koepf, Wolfram. "Algorithmic development of power series. Artificial intelligence and symbolic mathematical computing." Lecture Notes in Computer Science, Vol. 737, pp. 195-213. Edited by J. Calmet and J. A. Campbell.  Berlin-Heidelberg: Springer, 1993.

  

Koepf, Wolfram. "Examples for the algorithmic calculation of formal Puiseux, Laurent and power series." SIGSAM Bulletin Vol. 27, (1993): 20-32.

  

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. 169-175. New York: ACM, 2003.

  

Koepf, Wolfram. "Power series in computer algebra." Journal of Symbolic Computation, Vol. 13, (1992): 581-603.

  

van Hoeij, Mark. "Finite singularities and hypergeometric solutions of linear recurrence equations." J. Pure and Appl. Algebra, Vol. 139, (1999): 109-131.

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

 


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