Factor3
Factor3[exp]
factors a rational function exp
over the field of complex numbers.
Factor3
is primarily meant to be used on matrices from differential equations and Feynman parametric representations of loop integrals. Its main goal is to rewrite all denominators such, that they can be integrated in terms of HPLs or GPLs (when possible).
To avoid performance bottlenecks, in the case of rational functions only the denominator will be factored by default. This can be changed by setting the option Numerator
to True
.
See also
Overview , FCPartialFractionForm .
Examples
Factor3[ (1 - 4 x ) (1 + 3 y )]
− 12 ( x − 1 4 ) ( y + 1 3 ) -12 \left(x-\frac{1}{4}\right) \left(y+\frac{1}{3}\right) − 12 ( x − 4 1 ) ( y + 3 1 )
Factor3[ 16 * (1 - 2 * eps)^ 2 * x ^ 2 ]
64 ( eps − 1 2 ) 2 x 2 64 \left(\text{eps}-\frac{1}{2}\right)^2 x^2 64 ( eps − 2 1 ) 2 x 2
Factor3[ 2 * (32904490323 + 164521613783 * eps + 1256744 * eps^ 2 )* (11 - 5 * eps - 47 * eps^ 2 + 44 * eps^ 3 )]
110593472 ( eps + 1 264 ( 1 − i 3 ) − 137143 + 198 i 122615 3 + 2869 ( 1 + i 3 ) 264 − 137143 + 198 i 122615 3 − 47 132 ) ( eps + 1 264 ( 1 + i 3 ) − 137143 + 198 i 122615 3 + 2869 ( 1 − i 3 ) 264 − 137143 + 198 i 122615 3 − 47 132 ) ( eps − 628374 − 1570927 − 2467796558401 ) ( eps − 104729 ( − 1570927 − 2467796558401 ) 2513488 ) ( eps + 1 132 ( − 47 − 2869 − 137143 + 198 i 122615 3 − − 137143 + 198 i 122615 3 ) ) 110593472 \left(\text{eps}+\frac{1}{264} \left(1-i \sqrt{3}\right) \sqrt[3]{-137143+198 i \sqrt{122615}}+\frac{2869 \left(1+i \sqrt{3}\right)}{264 \sqrt[3]{-137143+198 i \sqrt{122615}}}-\frac{47}{132}\right) \left(\text{eps}+\frac{1}{264} \left(1+i \sqrt{3}\right) \sqrt[3]{-137143+198 i \sqrt{122615}}+\frac{2869 \left(1-i \sqrt{3}\right)}{264 \sqrt[3]{-137143+198 i \sqrt{122615}}}-\frac{47}{132}\right) \left(\text{eps}-\frac{628374}{-1570927-\sqrt{2467796558401}}\right) \left(\text{eps}-\frac{104729 \left(-1570927-\sqrt{2467796558401}\right)}{2513488}\right) \left(\text{eps}+\frac{1}{132} \left(-47-\frac{2869}{\sqrt[3]{-137143+198 i \sqrt{122615}}}-\sqrt[3]{-137143+198 i \sqrt{122615}}\right)\right) 110593472 ( eps + 264 1 ( 1 − i 3 ) 3 − 137143 + 198 i 122615 + 264 3 − 137143 + 198 i 122615 2869 ( 1 + i 3 ) − 132 47 ) ( eps + 264 1 ( 1 + i 3 ) 3 − 137143 + 198 i 122615 + 264 3 − 137143 + 198 i 122615 2869 ( 1 − i 3 ) − 132 47 ) ( eps − − 1570927 − 2467796558401 628374 ) ( eps − 2513488 104729 ( − 1570927 − 2467796558401 ) ) ( eps + 132 1 ( − 47 − 3 − 137143 + 198 i 122615 2869 − 3 − 137143 + 198 i 122615 ) )
mat = {{ (2 - 2 * eps)/ x , 0 , 0 , 0 , 0 }, { 0 , (2 - 2 * eps)/ (2 * x ), 0 , 0 , 0 },
{ 0 , (- 2 + 2 * eps)/ (x - 4 * x ^ 2 ), (6 - 2 * (4 - 2 * eps))/ (1 - 4 * x ), 0 , 0 },
{ (- 2 + 2 * eps)/ (x - 4 * x ^ 2 ), 0 , 0 , (2 - 2 * eps + 4 * (5 - 2 * (4 - 2 * eps))* x )/ (2 * (1 -
4 * x )* x ), 0 }, { (2 - 2 * eps)^ 2 / (16 * (1 - x )* x ^ 2 ), - 1 / 8 * (2 - 2 * eps)^ 2 / ((1 - x )* x ^ 2 ),
0 , 0 , - ((7 - 2 * (4 - 2 * eps) - 13 * x + 4 * (4 - 2 * eps)* x )/ (2 * x - 2 * x ^ 2 ))}} ;
( 2 − 2 eps x 0 0 0 0 0 2 − 2 eps 2 x 0 0 0 0 − 2 eps − 2 4 ( x − 1 4 ) x − 6 − 2 ( 4 − 2 eps ) 4 ( x − 1 4 ) 0 0 − 2 eps − 2 4 ( x − 1 4 ) x 0 0 − 4 ( 5 − 2 ( 4 − 2 eps ) ) x − 2 eps + 2 8 ( x − 1 4 ) x 0 − ( 2 − 2 eps ) 2 16 ( x − 1 ) x 2 ( 2 − 2 eps ) 2 8 ( x − 1 ) x 2 0 0 − − 4 ( 4 − 2 eps ) x + 2 ( 4 − 2 eps ) + 13 x − 7 2 ( x − 1 ) x ) \left(
\begin{array}{ccccc}
\frac{2-2 \;\text{eps}}{x} & 0 & 0 & 0 & 0 \\
0 & \frac{2-2 \;\text{eps}}{2 x} & 0 & 0 & 0 \\
0 & -\frac{2 \;\text{eps}-2}{4 \left(x-\frac{1}{4}\right) x} & -\frac{6-2 (4-2 \;\text{eps})}{4 \left(x-\frac{1}{4}\right)} & 0 & 0 \\
-\frac{2 \;\text{eps}-2}{4 \left(x-\frac{1}{4}\right) x} & 0 & 0 & -\frac{4 (5-2 (4-2 \;\text{eps})) x-2 \;\text{eps}+2}{8 \left(x-\frac{1}{4}\right) x} & 0 \\
-\frac{(2-2 \;\text{eps})^2}{16 (x-1) x^2} & \frac{(2-2 \;\text{eps})^2}{8 (x-1) x^2} & 0 & 0 & -\frac{-4 (4-2 \;\text{eps}) x+2 (4-2 \;\text{eps})+13 x-7}{2 (x-1) x} \\
\end{array}
\right) x 2 − 2 eps 0 0 − 4 ( x − 4 1 ) x 2 eps − 2 − 16 ( x − 1 ) x 2 ( 2 − 2 eps ) 2 0 2 x 2 − 2 eps − 4 ( x − 4 1 ) x 2 eps − 2 0 8 ( x − 1 ) x 2 ( 2 − 2 eps ) 2 0 0 − 4 ( x − 4 1 ) 6 − 2 ( 4 − 2 eps ) 0 0 0 0 0 − 8 ( x − 4 1 ) x 4 ( 5 − 2 ( 4 − 2 eps )) x − 2 eps + 2 0 0 0 0 0 − 2 ( x − 1 ) x − 4 ( 4 − 2 eps ) x + 2 ( 4 − 2 eps ) + 13 x − 7