check for the existence of a primitive element, and perform accurate integration - Maple Help

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LinearOperators[IntegrateSols] - check for the existence of a primitive element, and perform accurate integration

Calling Sequence

IntegrateSols(L, x, case)

Parameters

L

-

an Ore operator

x

-

the name of the independent variable

case

-

a parameter indicating the case of the equation ('differential' or 'shift')

Description

• 

The LinearOperators[IntegrateSols] function performs "accurate integration". That is, it solves the following problem. Let y satisfy L(y)=0 and g satisfy delta(g)=y, where delta means the usual derivative in the differential case and the first difference in the shift case. The routine builds an annihilator S for g of the same degree as that of L, and an operator K such that g=K(y) if both exist. Otherwise, it returns NULL.

• 

An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2x+xD+x+1D2+D3.

• 

There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

Examples

withLinearOperators:

expr:=xlogx2

expr:=xlnx2

(1)

An annihilator for expr is

L:=FactoredAnnihilatorexpr,x,'differential'

L:=FactoredOrePoly32x,1,12x,1,12x,1

(2)

which can be written in non-factored form as

L:=FactoredOrePolyToOrePolyL,x,'differential'

L:=OrePoly18x3,14x2,32x,1

(3)

IntegrateSolsL,x,'differential'

OrePoly278x3,134x2,32x,1,OrePoly2627x,49x2,827x3

(4)

See Also

DEtools/integrate_sols, LinearOperators, LinearOperators[converters], LinearOperators[FactoredAnnihilator], LinearOperators[FactoredOrePolyToOrePoly]

References

  

Abramov, S. A., and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transforms and Special Functions. (1999): 3-12.


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