Table 6 Fractional factorials and multifactorials
From: Factorials of real negative and imaginary numbers - A new perspective
Fractional factorials and multifactorials of real positive numbers | |||||
|---|---|---|---|---|---|
Fractional factorials | Multifactorials | ||||
z | z! | (0.5z)! | (1.5z)! | (2z)!! | (3z)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.886 | 0.626 | 1.085 | 1.253 | 1.534 |
1 | 1 | 0.5 | 1.5 | 2 | 3 |
1.5 | 1.329 | 0.469 | 2.442 | 3.759 | 6.907 |
2 | 2 | 0.5 | 4.5 | 8 | 18 |
2.5 | 3.323 | 0.587 | 9.158 | 18.799 | 51.805 |
3 | 6 | 0.75 | 20.25 | 48 | 162 |
Fractional factorials and multifactorials of real negative numbers | |||||
Fractional factorials | Multifactorials | ||||
z | (-z)! | (-0.5)z! | (-1.5z)! | (-2z)!! | (-3z)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.886i | 0.626i | 1.085 | 1.253i | 1.534i |
1 | -1 | -0.5 | -1.5 | -2 | -3 |
1.5 | -0.139i | -0.469i | -2.442 | -3.759i | -6.907i |
2 | 2 | 0.5 | 4.5 | 8 | 18 |
2.5 | 3.323i | 0.587 | 9.157 | 18.799i | 51.805i |
3 | -6 | -0.75 | -20.25 | -48 | -162 |
Fractional factorials and multifactorials of imaginary positive numbers | |||||
Fractional factorials | Multifactorials | ||||
z | (iz!) | (0.5iz)! | (1.5iz)! | (2iz)!! | (3iz)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.626 + 0.626i | 0.443 + 0.443i | 0.767 + 0.767i | 0.886 + 0.886i | 1.085 + 1.085i |
1 | i | 0.5i | 1.5i | 2i | 3i |
1.5 | -0.939 + 0.939i | -0.332 + 0.332i | -1.726 + 1.726i | -2.658 + 2.658i | -4.862 + 4.862i |
2 | -2 | -0.5 | -4.5 | -8 | -18 |
2.5 | -2.349-2.349i | -0.415-0.415i | -6.475-6.475i | -13.29-13.29i | 36.6-36.6i |
3 | 6i | -0.75i | -20.25i | -48i | -162i |
Fractional factorials and multifactorials of imaginary negative numbers | |||||
Fractional factorials | Multifactorials | ||||
z | (-iz)! | (-0.5iz)! | (-1.5iz)! | (-2iz)!! | (-3iz)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.626-0.626i | 0.443-0.443i | 0.767-0.767i | 0.886-0.886i | 1.085-1.085i |
1 | -i | -0.5i | -1.5i | -2i | -3i |
1.5 | -0.939-0.939i | -0.332-0.332i | -1.726-1.726i | -2.658-2.658i | -4.862-4.862i |
2 | -2 | -0.5 | -4.5 | -8 | -18 |
2.5 | -2.349 + 2.349i | -0.415 + 0.415i | -6.475 + 6.475i | -13.29 + 13.29i | -36.6 + 36.6i |
3 | 6i | 0.75i | 20.25i | 48i | 162i |