FCPartialFractionForm[n, {{f1,x-r1,p1},{f2,x-r2,p2}, ...}, x]
is a special way of representing sums of rational functions of
x given by n +
\frac{f_1}{[x-r_1]^p_1} + \frac{f_2}{[x-r_2]^p_2} + \ldots
It is inspired by the parfracform from Maple and its
usage in E. Panzer’s HyperInt for the integration of multiple
polylogarithms.
Use ToFCPartialFractionForm to convert the given
expression to this notation and FromFCPartialFractionForm
to return back to the usual representation.
Overview, ToFCPartialFractionForm, FromFCPartialFractionForm.
Apart[c + x^2/(x - 1), x]c+x+\frac{1}{x-1}+1
ex1 = ToFCPartialFractionForm[c + x^2/(x - 1), x]\text{FCPartialFractionForm}\left(c+x+1,\left( \begin{array}{cc} \{x-1,-1\} & 1 \\ \end{array} \right),x\right)
FromFCPartialFractionForm[ex1]c+x+\frac{1}{x-1}+1
ex2 = ToFCPartialFractionForm[(-64*(-1 + z^2))/(15*(1 + z^2 + z^4)), z]\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \left\{z-\sqrt[3]{-1},-1\right\} & -\frac{32}{15} \\ \left\{z+\sqrt[3]{-1},-1\right\} & \frac{32}{15} \\ \left\{z-(-1)^{2/3},-1\right\} & \frac{32}{15} \\ \left\{z+(-1)^{2/3},-1\right\} & -\frac{32}{15} \\ \end{array} \right),z\right)
FromFCPartialFractionForm[ex2]\frac{32}{15 \left(z+\sqrt[3]{-1}\right)}+\frac{32}{15 \left(z-(-1)^{2/3}\right)}-\frac{32}{15 \left(z+(-1)^{2/3}\right)}-\frac{32}{15 \left(z-\sqrt[3]{-1}\right)}
FromFCPartialFractionForm[ex2, Factoring -> Simplify]-\frac{64 \left(z^2-1\right)}{15 \left(z^4+z^2+1\right)}