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 ReduceHyperexp
 a reduction algorithm for hyperexponential functions

 Calling Sequence ReduceHyperexp(H, x, newH)

Parameters

 H - hyperexponential function of x H1 - hyperexponential function of x H2 - hyperexponential function of x x - variable newH - (optional) name; assigned a computed equivalence of H

Description

 • For a specified hyperexponential function H of x, the (H1, H2) := ReduceHyperexp(H, x, newH) calling sequence constructs two hyperexponential functions H1 and H2 such that $H\left(x\right)=\frac{{ⅆ}}{{ⅆ}x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{H1}\left(x\right)+\mathrm{H2}\left(x\right)$ and the certificate $\frac{\frac{{ⅆ}}{{ⅆ}x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{H2}\left(x\right)}{\mathrm{H2}\left(x\right)}$ has a differential rational normal form $r,s,u,v$ with v of minimal degree.
 • The output from ReduceHyperexp is a sequence of two elements $\mathrm{H1},\mathrm{H2}$ each of which is either $0$ or written in the form

$H\left(x\right)=V\left(x\right){ⅇ}^{{\int }F\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x}$

 (The form shown above is called a multiplicative decomposition of the hyperexponential function $H\left(x\right)$.)
 • ReduceHyperexp is a generalization of the reduction algorithm for rational functions by Hermite (recall that a rational function is also a hyperexponential function). It also covers the differential Gosper's algorithm.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $H≔\frac{\mathrm{exp}\left(\mathrm{Int}\left(\frac{2x-7}{{\left(x+4\right)}^{2}},x\right)\right)\left({x}^{6}+16{x}^{5}+103{x}^{4}+327{x}^{3}+647{x}^{2}+737x+194\right)}{{\left(x-1\right)}^{2}{\left(x+2\right)}^{4}{\left(x+4\right)}^{2}}$
 ${H}{≔}\frac{{{ⅇ}}^{{\int }\frac{{2}{}{x}{-}{7}}{{\left({x}{+}{4}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{}\left({{x}}^{{6}}{+}{16}{}{{x}}^{{5}}{+}{103}{}{{x}}^{{4}}{+}{327}{}{{x}}^{{3}}{+}{647}{}{{x}}^{{2}}{+}{737}{}{x}{+}{194}\right)}{{\left({x}{-}{1}\right)}^{{2}}{}{\left({x}{+}{2}\right)}^{{4}}{}{\left({x}{+}{4}\right)}^{{2}}}$ (1)
 > $\mathrm{H1},\mathrm{H2}≔\mathrm{ReduceHyperexp}\left(H,x,'\mathrm{nH}'\right)$
 ${\mathrm{H1}}{,}{\mathrm{H2}}{≔}{-}\frac{\left({24}{}{{x}}^{{3}}{+}{143}{}{{x}}^{{2}}{+}{292}{}{x}{+}{216}\right){}{{ⅇ}}^{{\int }{-}\frac{{15}}{{\left({x}{+}{4}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}}{\left({x}{-}{1}\right){}{\left({x}{+}{2}\right)}^{{3}}}{,}\frac{\left({{x}}^{{3}}{+}{17}{}{{x}}^{{2}}{+}{88}{}{x}{-}{231}\right){}{{ⅇ}}^{{\int }\frac{{-}{23}{-}{2}{}{x}}{{\left({x}{+}{4}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}}{{x}{-}{1}}$ (2)
 > $\mathrm{nH}$
 $\frac{\left({{x}}^{{6}}{+}{16}{}{{x}}^{{5}}{+}{103}{}{{x}}^{{4}}{+}{327}{}{{x}}^{{3}}{+}{647}{}{{x}}^{{2}}{+}{737}{}{x}{+}{194}\right){}{{ⅇ}}^{{\int }{-}\frac{{15}}{{\left({x}{+}{4}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}}{{\left({x}{-}{1}\right)}^{{2}}{}{\left({x}{+}{2}\right)}^{{4}}}$ (3)
 > $H≔-\frac{\mathrm{exp}\left(\mathrm{Int}\left(\frac{2x-7}{{\left(x+4\right)}^{2}},x\right)\right)\left({x}^{2}+27x+62\right)}{{\left(x+2\right)}^{4}{\left(x+4\right)}^{2}}$
 ${H}{≔}{-}\frac{{{ⅇ}}^{{\int }\frac{{2}{}{x}{-}{7}}{{\left({x}{+}{4}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{}\left({{x}}^{{2}}{+}{27}{}{x}{+}{62}\right)}{{\left({x}{+}{2}\right)}^{{4}}{}{\left({x}{+}{4}\right)}^{{2}}}$ (4)
 > $\mathrm{H1},\mathrm{H2}≔\mathrm{ReduceHyperexp}\left(H,x\right)$
 ${\mathrm{H1}}{,}{\mathrm{H2}}{≔}{-}\frac{\left({{x}}^{{2}}{+}{8}{}{x}{+}{16}\right){}{{ⅇ}}^{{\int }{-}\frac{{15}}{{\left({x}{+}{4}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}}{{\left({x}{+}{2}\right)}^{{3}}}{,}{0}$ (5)

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.