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.

$\mathrm{with}\left(\mathrm{genfunc}\right)\:$

$\mathrm{rgf\_pfrac}\left(\frac{1+z}{1-3z+2z^2}\,z\right)$

$-\frac{3}{2z-1}+\frac{2}{z-1}$

$\mathrm{rgf\_pfrac}\left(\frac{1}{1-z-z^2}\,z\right)$

$\frac{2\sqrt{5}}{5\left(2z+\sqrt{5}+1\right)}-\frac{2\sqrt{5}}{5\left(2z-\sqrt{5}+1\right)}$

$\mathrm{rgf\_pfrac}\left(\frac{1}{1-z-z^2-z^3}\,z\right)$

$\sum _{\mathrm{\_R}=\mathrm{RootOf}\left({\mathrm{\_Z}}^3+{\mathrm{\_Z}}^2+\mathrm{\_Z}-1\right)}\frac{\underset{z\to \mathrm{\_R}}{lim}\left(-\frac{z-\mathrm{\_R}}{z^3+z^2+z-1}\right)}{z-\mathrm{\_R}}$

$\mathrm{rgf\_pfrac}\left(\frac{1}{1-z-z^2-z^5+z^6}\,z\,'\mathrm{no\_RootOf}'\right)$

$\mathrm{Rgf\_pfrac}\left(\frac{1}{z^6-z^5-z^2-z+1}\,z\right)$