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 MultiplicativeDecomposition
 construct two multiplicative decompositions of a hyperexponential function

 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, $i=\left\{1,2\right\}$.
 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 $H\left(x\right)=V\left(x\right){ⅇ}^{{∫}F\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x}$. Let R be the rational certificate of H, i.e., $R=\frac{\frac{{ⅆ}}{{ⅆ}x}H\left(x\right)}{H\left(x\right)}$. 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 $\mathrm{RationalCanonicalForm}[i]\left(\frac{\frac{{\partial }}{{\partial }x}H}{H},x\right)$, for $i=1,2$.
 • The output is of the form $V\left(x\right){ⅇ}^{{∫}F\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x}$ where V and F are rational function of x over K.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $R≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^{2}}-\frac{9}{{\left(x-1\right)}^{2}}-\frac{9{x}^{2}+12}{{x}^{3}+4x-2}+\frac{1}{{\left({x}^{3}+4x-2\right)}^{2}}$
 ${R}{≔}\frac{{4}}{{x}{-}{2}}{+}\frac{{4}}{{x}{+}{1}}{-}\frac{{3}}{{\left({x}{+}{1}\right)}^{{2}}}{-}\frac{{9}}{{\left({x}{-}{1}\right)}^{{2}}}{-}\frac{{9}{}{{x}}^{{2}}{+}{12}}{{{x}}^{{3}}{+}{4}{}{x}{-}{2}}{+}\frac{{1}}{{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}$ (1)
 > $H≔{ⅇ}^{{∫}R\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x}$
 ${H}{≔}{{ⅇ}}^{{∫}\left(\frac{{4}}{{x}{-}{2}}{+}\frac{{4}}{{x}{+}{1}}{-}\frac{{3}}{{\left({x}{+}{1}\right)}^{{2}}}{-}\frac{{9}}{{\left({x}{-}{1}\right)}^{{2}}}{-}\frac{{9}{}{{x}}^{{2}}{+}{12}}{{{x}}^{{3}}{+}{4}{}{x}{-}{2}}{+}\frac{{1}}{{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (2)
 > $\mathrm{MultiplicativeDecomposition}[1]\left(H,x\right)$
 $\frac{{\left({x}{+}{1}\right)}^{{4}}{}{\left({x}{-}{2}\right)}^{{4}}{}{{ⅇ}}^{{∫}\frac{{-}{12}{}{{x}}^{{8}}{-}{12}{}{{x}}^{{7}}{-}{108}{}{{x}}^{{6}}{-}{48}{}{{x}}^{{5}}{-}{239}{}{{x}}^{{4}}{+}{48}{}{{x}}^{{3}}{-}{50}{}{{x}}^{{2}}{+}{144}{}{x}{-}{47}}{{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}}{{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{3}}}$ (3)
 > $\mathrm{MultiplicativeDecomposition}[2]\left(H,x\right)$
 ${\left({x}{-}{2}\right)}^{{4}}{}{{ⅇ}}^{{∫}\frac{{-}{5}{}{{x}}^{{9}}{-}{16}{}{{x}}^{{8}}{-}{14}{}{{x}}^{{7}}{-}{134}{}{{x}}^{{6}}{+}{39}{}{{x}}^{{5}}{-}{331}{}{{x}}^{{4}}{+}{96}{}{{x}}^{{3}}{+}{32}{}{{x}}^{{2}}{+}{16}{}{x}{-}{7}}{{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (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.