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 AreSimilar
 test if two hyperexponential functions are similar

 Calling Sequence AreSimilar(H1, H2, x)

Parameters

 H1 - hyperexponential function of x H2 - hyperexponential function of x x - variable

Description

 • Let $\mathrm{H1},\mathrm{H2}$ be hyperexponential functions of x over a field K of characteristic 0. The AreSimilar(H1,H2,x) command returns true if $\mathrm{H1}\left(x\right)$ and $\mathrm{H2}\left(x\right)$ are similar. Otherwise, it returns false.
 • H1 and H2 are similar if their ratio can be written as the product of a rational function and a constant in some extension of K.

Examples

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

References

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