Table 1 The Eq. (9) could yield numerous solutions for the function \(F\) based on the selected values for \({m}_{2}, {m}_{1}\) and \({m}_{0}\)62,63.
\({m}_{2}\) | \({m}_{1}\) | \({m}_{0}\) | F | |
|---|---|---|---|---|
1 | \({r}^{2}\) | \(-(1+{r}^{2})\) | 1 | \(sn, cd\) |
2 | \({r}^{2}\) | \(2{r}^{2}-1\) | \(1-{r}^{2}\) | \(cn\) |
3 | − 1 | \(2-{r}^{2}\) | \({r}^{2}-1\) | \(dn\) |
4 | 1 | \(-(1+{r}^{2})\) | \({r}^{2}\) | \(ns, dc\) |
5 | \(1-{r}^{2}\) | \(2{r}^{2}-1\) | \(-1\) | \(nd\) |
6 | \({r}^{2}-1\) | \(2-{r}^{2}\) | \(-1\) | \(nd\) |
7 | \(1-{r}^{2}\) | \(2-{r}^{2}\) | 1 | \(sc\) |
8 | \(-{r}^{2}(1-{r}^{2})\) | \(2{r}^{2}-1\) | 1 | \(sd\) |
9 | 1 | \(2-{r}^{2}\) | \(1-{r}^{2}\) | \(cs\) |
10 | 1 | \(2{r}^{2}-1\) | \({r}^{2}(1-{r}^{2})\) | \(ds\) |
11 | \(\frac{-1}{4}\) | \(\frac{{r}^{2}+1}{2}\) | \(\frac{-{(1-{r}^{2})}^{2}}{4}\) | \(rcn\mp dn\) |
12 | \(\frac{1}{4}\) | \(\frac{-2{r}^{2}+1}{2}\) | \(\frac{1}{4}\) | \(ns\mp cs\) |
13 | \(\frac{1-{r}^{2}}{4}\) | \(\frac{{r}^{2}+1}{2}\) | \(\frac{1-{r}^{2}}{4}\) | \(nc\mp sc\) |
14 | \(\frac{1}{4}\) | \(\frac{{r}^{2}-2}{2}\) | \(\frac{{r}^{4}}{4}\) | \(ns\mp ds\) |
15 | \(\frac{{r}^{2}}{4}\) | \(\frac{{r}^{2}-2}{2}\) | \(\frac{{r}^{2}}{4}\) | \(sn\mp icn, \frac{sn}{\sqrt{1-{r}^{2}sn\mp cn}}\) |
16 | \(\frac{1}{4}\) | \(\frac{1-2{r}^{2}}{2}\) | \(\frac{1}{4}\) | \(rcn\mp idn, \frac{sn}{1\mp cn}\) |
17 | \(\frac{{r}^{2}}{4}\) | \(\frac{{r}^{2}-2}{2}\) | \(\frac{1}{4}\) | \(\frac{sn}{1\mp dn}\) |
18 | \(\frac{{r}^{2}-1}{4}\) | \(\frac{{r}^{2}+1}{2}\) | \(\frac{{r}^{2}-1}{4}\) | \(\frac{dn}{1\mp rsn}\) |
19 | \(\frac{1-{r}^{2}}{4}\) | \(\frac{{r}^{2}+1}{2}\) | \(\frac{-{r}^{2}+1}{4}\) | \(\frac{cn}{1\mp sn}\) |
20 | \(\frac{{(1-{r}^{2})}^{2}}{4}\) | \(\frac{{r}^{2}+1}{2}\) | \(\frac{1}{4}\) | \(\frac{sn}{dn\mp cn}\) |
21 | \(\frac{{r}^{4}}{4}\) | \(\frac{{r}^{2}-2}{2}\) | \(\frac{1}{4}\) | \(\frac{cn}{\sqrt{1-{r}^{2}\mp dn}}\) |
