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genfunc

 rgf_pfrac
 partial fractions over the complex numbers

 Calling Sequence rgf_pfrac(Fz, z) rgf_pfrac(Fz, z, 'no_RootOf')

Parameters

 Fz - rational generating function z - name, generating function variable

Description

 • Computes the complete partial fraction expansion of Fz over the complex numbers.
 • The denominator of Fz is factored using factor. Any factors that are polynomials of degree 2 are then factored over the complex numbers. Any factors that are polynomials of degree greater than 2 are represented in factored form using Sum and RootOf expressions.
 • If the optional argument 'no_RootOf' is used, the denominator will be completely factored over the complex numbers. If the denominator cannot be factored, an inert Pfrac expression is returned.
 • The global variables _J and _R are used in the RootOf expressions.
 • The command with(genfunc,rgf_pfrac) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{genfunc}\right):$
 > $\mathrm{rgf_pfrac}\left(\frac{1+z}{1-3z+2{z}^{2}},z\right)$
 $\frac{{2}}{{z}{-}{1}}{-}\frac{{3}}{{2}{}{z}{-}{1}}$ (1)
 > $\mathrm{rgf_pfrac}\left(\frac{1}{1-z-{z}^{2}},z\right)$
 $\frac{{2}}{{5}}{}\frac{\sqrt{{5}}}{{2}{}{z}{+}{1}{+}\sqrt{{5}}}{-}\frac{{2}}{{5}}{}\frac{\sqrt{{5}}}{{2}{}{z}{-}\sqrt{{5}}{+}{1}}$ (2)
 > $\mathrm{rgf_pfrac}\left(\frac{1}{1-z-{z}^{2}-{z}^{3}},z\right)$
 ${\sum }_{{\mathrm{_R}}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{+}{{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{-}{1}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\underset{{z}{→}{\mathrm{_R}}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({-}\frac{{z}{-}{\mathrm{_R}}}{{{z}}^{{3}}{+}{{z}}^{{2}}{+}{z}{-}{1}}\right)}{{z}{-}{\mathrm{_R}}}$ (3)
 > $\mathrm{rgf_pfrac}\left(\frac{1}{1-z-{z}^{2}-{z}^{5}+{z}^{6}},z,'\mathrm{no_RootOf}'\right)$
 ${\mathrm{Rgf_pfrac}}{}\left(\frac{{1}}{{{z}}^{{6}}{-}{{z}}^{{5}}{-}{{z}}^{{2}}{-}{z}{+}{1}}{,}{z}\right)$ (4)