椭圆函数的雅可比高次代数分式变换

青青子衿

\begin{gather*}
\quad\begin{split}
k'&=\sqrt{1-k^2}\\
l'&=\sqrt{1-l^2}\\
\ell'&=\sqrt{1-\ell^2}\\
K=K(k)&=\int_0^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k^2t^2)}}\\
K'=K(k')&=\int_0^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k'^2t^2)}}\\
\\
\end{split}\\
\frac{K(k')}{K(k)}=p\cdot\frac{K(l')}{K(l)}=\frac{1}{p}\cdot\frac{K(\ell')}{K(\ell)}\\
\\
\begin{aligned}
M=\frac{K(k)}{K(l)}&&
\hat{M}=(-1)^{\frac{p-1}{2}}\frac{K(\ell')}{K(k')}\\
\end{aligned}
\end{gather*}

\begin{gather*}
\quad\begin{aligned}
\ell&=k^p\prod_{s=1}^{\frac{p-1}{2}}\operatorname{sn}^4\left(K-s\frac{2K}{p},k\right)&\hat{M}&=(-1)^{\frac{p-1}{2}}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{\operatorname{sn}^2(K-s\frac{2K}{p},k)}{\operatorname{sn}^2(s\frac{2K}{p},k)}\right)\\
&=k^p\prod_{s=1}^{\frac{p-1}{2}}\operatorname{cd}^4\left(s\frac{2K}{p},k\right)&&=(-1)^{\frac{p-1}{2}}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{\operatorname{cs}^2(s\frac{2K}{p},k)}{\operatorname{dn}^2(s\frac{2K}{p},k)}\right)\\
l&=k^{2-p}\prod_{s=1}^{\frac{p-1}{2}}\operatorname{dn}^4\left(K'-s\frac{2K'}{p},k'\right)&M&=\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{\operatorname{sn}^2(s\frac{2K'}{p},k')}{\operatorname{sn}^2(K'-s\frac{2K'}{p},k')}\right)\\
&=k^p\prod_{s=1}^{\frac{p-1}{2}}\operatorname{nd}^4\left(s\frac{2K'}{p},k'\right)&&=\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{\operatorname{dn}^2(s\frac{2K'}{p},k')}{\operatorname{cs}^2(s\frac{2K'}{p},k')}\right)\\

\end{aligned}
\end{gather*}

\begin{align*}
\qquad\operatorname{sn}\left(\frac{u}{\hat{M}},\ell\right)
&=\frac{\operatorname{sn}\left(u,k\right)}{\hat{M}}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}^2(u,k)}{\operatorname{sn}^2(s\frac{2K}{p},k)}}{1-\frac{k^2\operatorname{sn}^2(u,k)}{\operatorname{ns}^2(s\frac{2K}{p},k)}}\right)\\

\operatorname{sn}\left(M\cdot{u},l\right)
&=M\cdot\operatorname{sn}\left(u,k\right)\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1+\frac{\operatorname{sn}^2(u,k)}{\operatorname{sc}^2(s\frac{2K'}{p},k')}}{1+\frac{k^2\operatorname{sn}^2(u,k)}{\operatorname{cs}^2(s\frac{2K'}{p},k')}}\right)\\

\end{align*}

1. pa = 11;
2. ka = Sqrt[ModularLambda[I*Sqrt[pa]]];
3. la = Sqrt[ModularLambda[I/Sqrt[pa]]];
4. \[ScriptL]a = Sqrt[ModularLambda[I*pa*Sqrt[pa]]];
5. N[{EllipticK[1 - ka^2]/EllipticK[ka^2],
6.    pa*EllipticK[1 - la^2]/EllipticK[la^2],
7.    1/pa*EllipticK[1 - \[ScriptL]a^2]/EllipticK[\[ScriptL]a^2]},
8.   20] // Column
10. N[{ka^pa*
11.     Product[JacobiSN[EllipticK[ka^2] - s (2 EllipticK[ka^2])/pa,
12.        ka^2]^4, {s, 1, (pa - 1)/2}],
13.    ka^pa*Product[
14.      JacobiCD[s (2 EllipticK[ka^2])/pa, ka^2]^4, {s, 1, (pa - 1)/2}],
15.    Sqrt[ModularLambda[I*pa*Sqrt[pa]]]}, 20] // Column
17. N[{ka^(2 - pa) Product[
18.      JacobiDN[EllipticK[1 - ka^2] - s (2 EllipticK[1 - ka^2])/pa,
19.        1 - ka^2]^4, {s, 1, (pa - 1)/2}],
20.    ka^pa*Product[
21.      JacobiND[(2 s*EllipticK[1 - ka^2])/pa, 1 - ka^2]^4, {s,
22.       1, (pa - 1)/2}],
23.    Sqrt[ModularLambda[I/Sqrt[pa]]]}, 20] // Column
25. pa3 = 3;
26. ka3 = Sqrt[ModularLambda[I Sqrt[pa3]]];
27. \[ScriptL]a3 =
28.   Sqrt[ModularLambda[
29.     I*pa3*Sqrt[
30.       pa3]]];  (*(2^(7/6)*3+ 2^(17/6)*3^(1/2)-2^(1/2)*3^2- \
31. 2^(7/6)*3^(1/2)-2^(1/2)*3^(1/2))/12*)
32. N[(Sqrt[3] x (1 + (2 Sqrt[3] - 3)/6 x^2))/(1 + Sqrt[3]/2 x^2), 20]
33. N[Sqrt[3] x*
34.   Product[(1 +
35.       x^2/JacobiSC[s (2 EllipticK[1 - ka3^2])/pa3,
36.          1 - ka3^2]^2)/(1 + (ka3^2*x^2)/
37.        JacobiCS[s (2 EllipticK[1 - ka3^2])/pa3, 1 - ka3^2]^2), {s,
38.     1, (pa3 - 1)/2}], 20]
39. N[(-(((1 + 2^(1/3))^2 3^(1/2))/3)
40.     x (1 - ((
41.       3^(1/4) - 2^(1/3)*3^(1/4) + 2^(2/3)*3^(1/4) + 3^(3/4) +
42.        2^(1/3) *3^(3/4))/6)^2 x^2))/(
43. 1 - ((2^(1/6)*3^(3/4) + 2^(1/2) *3^(3/4) - 2^(1/6)*3^(5/4))/
44.     6)^2 x^2), 20]
45. N[-(((1 + 2^(1/3))^2 3^(1/2))/3) x*
46.   Product[(1 - x^2/JacobiSN[(2 s EllipticK[ka3^2])/pa3, ka3^2]^2)/(
47.    1 - (ka3^2 x^2)/JacobiNS[(2 s EllipticK[ka3^2])/pa3, ka3^2]^2), {s,
48.      1, (pa3 - 1)/2}], 20]
50. pb = 5;
51. kb = Sqrt[ModularLambda[I Sqrt[pb]]];
52. N[(Sqrt[5]
53.     x (1 + (Sqrt[2 + 2 Sqrt[5]] - 2 )/2  x^2 + (
54.       15 - Sqrt[5] - 5 Sqrt[2 + 2 Sqrt[5]])/20 x^4))/(
55.   1 + Sqrt[10 Sqrt[5] - 10]/
56.     2  x^2 + (-5 + 3 Sqrt[5] - Sqrt[50 Sqrt[5] - 110])/4 x^4),
57.   20] // Factor
58. N[Sqrt[5] x*
59.    Product[(
60.     1 + x^2/JacobiSC[s (2 EllipticK[1 - kb^2])/pb, 1 - kb^2]^2)/(
61.     1 + (kb^2 x^2)/
62.      JacobiCS[s (2 EllipticK[1 - kb^2])/pb, 1 - kb^2]^2), {s, 1, (
63.      pb - 1)/2}], 20] // Factor

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sci-hub.ru/10.1007/s00407-013-0131-3
sci-hub.ru/https://royalsocietypublishing.org/doi/10.1098/rstl.1874.0011

1. pa5 = 5;
2. ka5 = Sqrt[
3.    ModularLambda[
4.     I*Sqrt[pa5]]];  (*(Sqrt[2]-Sqrt[10]+2Sqrt[Sqrt[5]-1 ])/4*)
5. la5 = Sqrt[
6.    ModularLambda[
7.     I/Sqrt[pa5]]];  (*(Sqrt[10]-Sqrt[2]+2Sqrt[Sqrt[5]-1 ])/4*)
8. N[(Sqrt[5]
9.     x (1 + (Sqrt[2 + 2 Sqrt[5]] - 2)/2  x^2 + (
10.       15 - Sqrt[5] - 5 Sqrt[2 + 2 Sqrt[5]])/20 x^4))/(
11.   1 + Sqrt[10 Sqrt[5] - 10]/2  x^2 + (
12.     3 Sqrt[5] - 5 - Sqrt[50 Sqrt[5] - 110])/4 x^4), 20] // Factor
13. N[Sqrt[5] x*
14.    Product[(1 +
15.        x^2/JacobiSC[s (2 EllipticK[1 - ka5^2])/pa5, 1 - ka5^2]^2)/(
16.       1 + (ka5^2 x^2)/
17.         JacobiCS[s (2 EllipticK[1 - ka5^2])/pa5, 1 - ka5^2]^2), {s,
18.      1, (pa5 - 1)/2}], 20] // Factor
19. N[JacobiSN[Sqrt[5] EllipticF[ArcSin[x], ka5^2], la5^2] /.
20.   x -> 1/10, 20]
21. N[JacobiSN[Sqrt[5] InverseJacobiSN[x, ka5^2], la5^2] /.
22.   x -> 1/10, 20]
23. N[(Sqrt[5]
24.     x (1 + (Sqrt[2 + 2 Sqrt[5]] - 2)/2  x^2 + (
25.       15 - Sqrt[5] - 5 Sqrt[2 + 2 Sqrt[5]])/20 x^4))/(
26.   1 + Sqrt[10 Sqrt[5] - 10]/2  x^2 + (
27.     3 Sqrt[5] - 5 - Sqrt[50 Sqrt[5] - 110])/4 x^4) /. x -> 1/10, 20]

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青青子衿

\begin{align*}
\begin{split}
\frac{1-\operatorname{sn}\left(\frac{u}{\overline{M}},\ell\right)}{1+\operatorname{sn}\left(\frac{u}{\overline{M}},\ell\right)}
&=\frac{1-\operatorname{sn}\left(u,k\right)}{1+\operatorname{sn}\left(u,k\right)}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}(u,k)}{\operatorname{cd}(s\frac{4K}{p},k)}}{1+\frac{\operatorname{sn}(u,k)}{\operatorname{cd}(s\frac{4K}{p},k)}}\right)^2\\

\frac{1-\operatorname{sn}\left(M\cdot{u},l\right)}{1+\operatorname{sn}\left(M\cdot{u},l\right)}
&=\frac{1-\operatorname{sn}\left(u,k\right)}{1+\operatorname{sn}\left(u,k\right)}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}(u,k)}{\operatorname{nd}(s\frac{2K'}{p},k')}}{1+\frac{\operatorname{sn}(u,k)}{\operatorname{nd}(s\frac{2K'}{p},k')}}\right)^2
\end{split}
\end{align*}

1. pb = 5;
2. kb = Sqrt[ModularLambda[I Sqrt[pb]]];
3. \[ScriptL]b = Sqrt[ModularLambda[I*pb* Sqrt[pb]]];
4. lb = Sqrt[ModularLambda[I/ Sqrt[pb]]];
5. N[(-1)^((pb - 1)/2)
6.     JacobiSN[
7.     EllipticK[1 - \[ScriptL]b^2]/
8.      EllipticK[1 - kb^2] InverseJacobiSN[x, kb^2], \[ScriptL]b^2] /.
9.   x -> 1/10, 50]
10. N[(-1)^((pb - 1)/2)
11.     Product[JacobiDN[(2 s EllipticK[kb^2])/pb, kb^2]^2/
12.     JacobiCS[(2 s EllipticK[kb^2])/pb, kb^2]^2, {s, 1, (pb - 1)/2}]*
13.    x*Product[(1 - x^2/JacobiSN[(2 s EllipticK[kb^2])/pb, kb^2]^2)/(
14.     1 - (kb^2 x^2)/JacobiNS[(2 s EllipticK[kb^2])/pb, kb^2]^2), {s,
15.      1, (pb - 1)/2}] /. x -> 1/10, 50]
16. N[(1 - ((1 - x)/(1 + x))*\[Eta])/(1 + ((1 - x)/(1 +
17.            x))*\[Eta]) /. \[Eta] ->
18.     Product[(1 - x/JacobiCD[(4 s EllipticK[kb^2])/pb, kb^2])^2/(1 + x/
19.        JacobiCD[(4 s EllipticK[kb^2])/pb, kb^2])^2, {s,
20.       1, (pb - 1)/2}] /. x -> 1/10, 50]
21. N[JacobiSN[EllipticK[lb^2]/EllipticK[kb^2] InverseJacobiSN[x, kb^2],
22.    lb^2] /. x -> 1/10, 50]
23. N[Product[JacobiDN[(2 s EllipticK[1 - kb^2])/pb, 1 - kb^2]^2/
24.     JacobiCS[(2 s EllipticK[1 - kb^2])/pb, 1 - kb^2]^2, {s,
25.      1, (pb - 1)/2}] x*
26.    Product[(1 +
27.        x^2/JacobiSC[s (2 EllipticK[1 - kb^2])/pb,
28.           1 - kb^2]^2)/(1 + (kb^2 x^2)/
29.         JacobiCS[s (2 EllipticK[1 - kb^2])/pb, 1 - kb^2]^2), {s,
30.      1, (pb - 1)/2}] /. x -> 1/10, 50]
31. N[(1 - ((1 - x)/(1 + x))*\[Eta])/(1 + ((1 - x)/(1 +
32.            x))*\[Eta]) /. \[Eta] ->
33.     Product[((1 -
34.           x/JacobiND[s (2 EllipticK[1 - kb^2])/pb, 1 - kb^2])/(1 +
35.           x/JacobiND[s (2 EllipticK[1 - kb^2])/pb, 1 - kb^2]))^2, {s,
36.       1, (pb - 1)/2}] /. x -> 1/10, 50]
37. Clear["Global`*"]

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青青子衿

青青子衿 发表于 2025-1-19 20:25
\begin{align*}
\begin{split}
\frac{1-\operatorname{sn}\left(\frac{u}{\overline{M}},\ell\right)}{1+\operatorname{sn}\left(\frac{u}{\overline{M}},\ell\right)}
&=\frac{1-\operatorname{sn}\left(u,k\right)}{1+\operatorname{sn}\left(u,k\right)}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}(u,k)}{\operatorname{cd}(s\frac{4K}{p},k)}}{1+\frac{\operatorname{sn}(u,k)}{\operatorname{cd}(s\frac{4K}{p},k)}}\right)^2\\

\frac{1-\operatorname{sn}\left(M\cdot{u},l\right)}{1+\operatorname{sn}\left(M\cdot{u},l\right)}
&=\frac{1-\operatorname{sn}\left(u,k\right)}{1+\operatorname{sn}\left(u,k\right)}\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}(u,k)}{\operatorname{nd}(s\frac{2K'}{p},k')}}{1+\frac{\operatorname{sn}(u,k)}{\operatorname{nd}(s\frac{2K'}{p},k')}}\right)^2
\end{split}
\end{align*}

\begin{align*}
\begin{split}
\operatorname{cn}\left(\frac{u}{\,\overline{M}\,},\ell\right)
&=\operatorname{cn}\left(u,k\right)\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}^2(u,k)}{\operatorname{cd}^2(s\frac{2K}{p},k)}}{1-\frac{k^2\operatorname{sn}^2(u,k)}{\operatorname{ns}^2(s\frac{2K}{p},k)}}\right)\\

\operatorname{cn}\left(M\cdot{u},l\right)
&=\operatorname{cn}\left(u,k\right)\prod_{s=1}^{\frac{p-1}{2}}\left(\frac{1-\frac{\operatorname{sn}^2(u,k)}{\operatorname{nd}^2(s\frac{2K'}{p},k')}}{1+\frac{k^2\operatorname{sn}^2(u,k)}{\operatorname{cs}^2(s\frac{2K'}{p},k')}}\right)
\end{split}
\end{align*}

1. pb = 7;
2. kb = Sqrt[ModularLambda[I Sqrt[pb]]];
3. \[ScriptL]b = Sqrt[ModularLambda[I*pb*Sqrt[pb]]];
4. lb = Sqrt[ModularLambda[I/Sqrt[pb]]];
5. N[ JacobiCN[
6.    EllipticK[1 - \[ScriptL]b^2]/EllipticK[1 - kb^2] InverseJacobiSN[x,
7.       kb^2], \[ScriptL]b^2] /. x -> 1/10, 50]
8. N[Sqrt[1 - x^2]*
9.    Product[(1 - x^2/JacobiCD[(2 s EllipticK[kb^2])/pb, kb^2]^2)/(
10.     1 - (kb^2 x^2)/JacobiNS[(2 s EllipticK[kb^2])/pb, kb^2]^2), {s,
11.      1, (pb - 1)/2}] /. x -> 1/10, 50]
13. N[JacobiCN[EllipticK[lb^2]/EllipticK[kb^2] InverseJacobiSN[x, kb^2],
14.    lb^2] /. x -> 1/10, 50]
15. N[ Sqrt[1 - x^2]*
16.    Product[(
17.     1 - x^2/JacobiND[(2 s EllipticK[1 - kb^2])/pb, 1 - kb^2]^2)/(
18.     1 + (kb^2 x^2)/
19.      JacobiCS[(2 s EllipticK[1 - kb^2])/pb, 1 - kb^2]^2), {s,
20.      1, (pb - 1)/2}] /. x -> 1/10, 50]
21. Clear["Global`*"]

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