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 IsHyperexponential
 test if a given expression is a hyperexponential function

 Calling Sequence IsHyperexponential(H, x)

Parameters

 H - Maple expression x - variable

Description

 • The IsHyperexponential(H,x) command returns true if $H$ is a hyperexponential function of x. Otherwise, it returns false.
 A function H is hyperexponential of x if $\frac{\frac{{ⅆ}}{{ⅆ}x}H\left(x\right)}{H\left(x\right)}=R\left(x\right)$, a rational function of x. $R\left(x\right)$ is the certificate of $H\left(x\right)$. If the third optional argument is included, it is assigned the certificate $R\left(x\right)$.

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)
 > $\mathrm{IsHyperexponential}\left(H,x,'R'\right)$
 ${\mathrm{true}}$ (2)
 > $R$
 ${-}\frac{{25}{}{{x}}^{{8}}{+}{473}{}{{x}}^{{7}}{+}{3748}{}{{x}}^{{6}}{+}{16441}{}{{x}}^{{5}}{+}{46985}{}{{x}}^{{4}}{+}{87685}{}{{x}}^{{3}}{+}{89797}{}{{x}}^{{2}}{+}{40832}{}{x}{+}{17764}}{\left({{x}}^{{6}}{+}{16}{}{{x}}^{{5}}{+}{103}{}{{x}}^{{4}}{+}{327}{}{{x}}^{{3}}{+}{647}{}{{x}}^{{2}}{+}{737}{}{x}{+}{194}\right){}\left({x}{+}{2}\right){}\left({x}{-}{1}\right){}{\left({x}{+}{4}\right)}^{{2}}}$ (3)

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