DEtools
MultiplicativeDecomposition
construct two multiplicative decompositions of a hyperexponential function
Calling Sequence
Parameters
Description
Examples
References
MultiplicativeDecomposition[1](H, x)
MultiplicativeDecomposition[2](H, x)
H
-
hyperexponential function of x
x
variable
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 RationalCanonicalFormiⅆHⅆxH,x, for i=1,2.
The output is of the form Vxⅇ∫Fxⅆx where V and F are rational function of x over K.
withDEtools:
R≔4x−2+4x+1−3x+12−9x−12−9x2+12x3+4x−2+1x3+4x−22
H≔expIntR,x
H≔ⅇ∫4x−2+4x+1−3x+12−9x−12−9x2+12x3+4x−2+1x3+4x−22ⅆx
MultiplicativeDecomposition1H,x
x+14x−24ⅇ∫−12x8−12x7−108x6−48x5−239x4+48x3−50x2+144x−47x+12x−12x3+4x−22ⅆxx3+4x−23
MultiplicativeDecomposition2H,x
x−24ⅇ∫−5x9−16x8−14x7−134x6+39x5−331x4+96x3+32x2+16x−7x+12x−12x3+4x−22ⅆx
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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