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MultiplicativeDecomposition

  

construct two multiplicative decompositions of a hyperexponential function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MultiplicativeDecomposition[1](H, x)

MultiplicativeDecomposition[2](H, x)

Parameters

H

-

hyperexponential function of x

x

-

variable

Description

• 

Let H be a hyperexponential function of x over a field K of characteristic 0. The MultiplicativeDecomposition[i](H,x) calling sequence constructs the ith multiplicative decomposition for H, i=1,2.

  

If the MultiplicativeDecomposition command is called without an index, the first multiplicative decomposition is constructed.

• 

A multiplicative decomposition of H is a pair of rational functions F,V such that Hx=Vxⅇ∫Fxⅆx. Let R be the rational certificate of H, i.e., R=ⅆⅆxHxHx. Let F,V be a differential rational normal form of R. Then F,V is a multiplicative decomposition of H. Hence, each differential rational normal form F,V of the certificate R of H is also a multiplicative decomposition of H.

• 

The construction of MultiplicativeDecomposition[i](H,x) is based on RationalCanonicalForm[i]xHH,x, for i=1,2.

• 

The output is of the form Vxⅇ∫Fxⅆx where V and F are rational function of x over K.

Examples

withDEtools:

R4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22

R:=4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22

(1)

Hⅇ∫Rⅆx

H:=ⅇ∫4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22ⅆx

(2)

MultiplicativeDecomposition[1]H,x

x+14x24ⅇ∫12x812x7108x648x5239x4+48x350x2+144x47x+12x12x3+4x22ⅆxx3+4x23

(3)

MultiplicativeDecomposition[2]H,x

x24ⅇ∫5x916x814x7134x6+39x5331x4+96x3+32x2+16x7x+12x12x3+4x22ⅆx

(4)

References

  

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press, (2004): 183-190.

See Also

DEtools[AreSimilar]

DEtools[RationalCanonicalForm]

DEtools[ReduceHyperexp]

SumTools[Hypergeometric][MultiplicativeDecomposition]

 


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