construct two multiplicative decompositions of a hyperexponential function
hyperexponential function of x
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 H⁡x=V⁡x⁢ⅇ∫F⁡xⅆx. Let R be the rational certificate of H, i.e., R=ⅆⅆx⁢H⁡xH⁡x. 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]⁡∂∂x⁢HH,x, for i=1,2.
The output is of the form V⁡x⁢ⅇ∫F⁡xⅆx where V and F are rational function of x over K.
R ≔ 4x−2+4x+1−3x+12−9x−12−9⁢x2+12x3+4⁢x−2+1x3+4⁢x−22
H ≔ ⅇ∫Rⅆx
H ≔ ⅇ∫4x−2+4x+1−3x+12−9x−12−9⁢x2+12x3+4⁢x−2+1x3+4⁢x−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.
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