勒让德型椭圆曲线与j不变量

青青子衿

\begin{align*}
j(\tau)&=\frac{2^8 \left(k^2(\tau)+\left(1-k^2(\tau)\right)^2\right)^3}{k^4(\tau)\left(1-k^2(\tau)\right)^2}\\
|j(\tau)-1728|&=\left(\frac{2^3(1 +k^2(\tau))(2 -k^2(\tau)) (1 -2k^2(\tau))}{k^2(\tau)(1 -k^2(\tau))}\right)^2\\
j(\tau)&=\left(\left(\frac{{\eta}\left(\tau\right)}{{\eta}\left(2\tau\right)}\right)^8+2^8\left(\frac{{\eta}\left(2\tau\right)}{{\eta}\left(\tau\right)}\right)^{16}\right)^{3}\\
j(2\tau)&=\left(\left(\frac{{\eta}\left(\tau\right)}{{\eta}\left(2\tau\right)}\right)^{16}+2^4\left(\frac{{\eta}\left(2\tau\right)}{{\eta}\left(\tau\right)}\right)^8\right)^{3}\\
\\
k^2(\tau)&=\lambda\left(\tau\right)=2^4\left(\frac{{\eta}^{2}\left(2\tau\right)\eta\left(%
\tfrac{1}{2}\tau\right)}{{\eta}^{3}\left(\tau\right)}\right)^{8}\\


\frac{k^4\left(\tau\right)}{1-k^2\left(\tau\right)}&=2^8\left(\frac{\eta \left(2\tau\right)}{\eta \left(\tau\right)}\right)^{24}

\end{align*}

\begin{align*}
k(2\tau)&=\frac{2\big(1-\sqrt{1-k^2(\tau)}\,\big)}{k^2(\tau)}-1\\
&=\frac{1-\sqrt{1-k^2(\tau)}}{1+\sqrt{1-k^2(\tau)}}\\
k\Big(\frac{\tau}{2}\Big)&=\frac{2\sqrt{k(\tau)}}{1+k(\tau)}

\end{align*}



\begin{align*}
&16k^{2}+24kT^{2}+24k^{3}T^{2}+48k^{2}T^{2}\\
&\qquad+T^{4}+4kT^{4}+6k^{2}T^{4}+4k^{3}T^{4}+k^{4}T^{4}\\
&\quad\qquad-8\sqrt{k}(1+k)T\left(4-4k+4k^{2}-3T^{2}+10kT^{2}-3k^{2}T^{2}\right)=0
\end{align*}

\begin{align*}
k_{11}&=\frac{1}{6} \sqrt{18-6 \sqrt{24 (1-\sqrt{3}\,i) \sqrt[3]{35099+21 \sqrt{33}}+24 (1+\sqrt{3}\,i) \sqrt[3]{35099-21 \sqrt{33}}-1563}}\\
\end{align*}

1. N[((t + 6)^3 (t^3 + 18 t^2 + 84 t + 24)^3)/(t (t + 8)^3 (t + 9)^2) /.
2.   t -> 2^3*3^2 (DedekindEta[6 Sqrt[2] I]^5 DedekindEta[2 Sqrt[2] I])/(
3.     DedekindEta[3 Sqrt[2] I] DedekindEta[Sqrt[2] I]^5), 10]
4. N[((t + 12)^3 (t^3 + 252 t^2 + 3888 t + 15552)^3)/(
5.   t^6 (t + 8)^2 (t + 9)^3) /.
6.   t -> (DedekindEta[Sqrt[2] I]^5 DedekindEta[3 Sqrt[2] I])/(
7.    DedekindEta[2 Sqrt[2] I] DedekindEta[6 Sqrt[2] I]^5), 10]
8. N[1728 KleinInvariantJ[Sqrt[2] I], 10]
9. N[((t + 12)^3 (t^3 + 252 t^2 + 3888 t + 15552)^3)/(
10.   t^6 (t + 8)^2 (t + 9)^3) /.
11.   t -> 2^3*3^2 (DedekindEta[6 Sqrt[2] I]^5 DedekindEta[2 Sqrt[2] I])/(
12.     DedekindEta[3 Sqrt[2] I] DedekindEta[Sqrt[2] I]^5), 10]
13. N[((t + 6)^3 (t^3 + 18 t^2 + 84 t + 24)^3)/(t (t + 8)^3 (t + 9)^2) /.
14.   t -> (DedekindEta[Sqrt[2] I]^5 DedekindEta[3 Sqrt[2] I])/(
15.    DedekindEta[2 Sqrt[2] I] DedekindEta[6 Sqrt[2] I]^5), 10]
16. N[1728 KleinInvariantJ[6 Sqrt[2] I], 10]

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青青子衿

\begin{align*}
k\big(\tau=\sqrt{11}\,i\big)&=\frac{3\sqrt{2}+\sqrt{22}}{12}\\
&\qquad-\frac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}\\
&\qquad\qquad-\frac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\\
k\Big(\tau=\frac{\sqrt{11}}{\,\,11}i\Big)&=\frac{\sqrt{22}-3 \sqrt{2}}{12}\\
&\qquad+\frac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}\\
&\qquad\qquad+\frac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\\

k\big(\tau=1+\sqrt{11}\,i\big)&=-\frac{(7 \sqrt{11}-22)i}{3}\\
&\qquad+\frac{4i(4+\sqrt{33}-3 \sqrt{3}-\sqrt{11})(3 \sqrt{33}-17)^{1/3}}{3}\\
&\qquad\qquad-\frac{4i(4+3 \sqrt{3}-\sqrt{11}-\sqrt{33})(17+3 \sqrt{33})^{1/3}}{3}\\
k\Big(\tau=\dfrac{-1+\sqrt{11}\,i}{12}\Big)&=-\frac{17 \sqrt{2}-5 \sqrt{22}}{3}\\
&\qquad+\frac{(19 \sqrt{6}+7 \sqrt{22}-19 \sqrt{2}-5 \sqrt{66}\>\>\!\!)(3 \sqrt{33}-17)^{1/3}}{6}\\
&\qquad\quad-\frac{(7 \sqrt{22}+5 \sqrt{66}-19 \sqrt{2}-19 \sqrt{6}\>\!)(17+3 \sqrt{33}\>\!)^{1/3}}{6}\\

k\Big(\tau=\dfrac{1+\sqrt{11}\,i}{2}\Big)&=\frac{\sqrt{11}+5 i}{6} \\
&\qquad+\frac{(\sqrt{3}+\sqrt{11}-7 i-i \sqrt{33}\>\>\!\!)(3 \sqrt{33}-17)^{1/3}}{6}\\
&\qquad\qquad-\frac{(\sqrt{11}+i \sqrt{33}-7 i-\sqrt{3}\>\>\!\!)(17+3 \sqrt{33}\>\>\!\!)^{1/3}}{6}\\
k\Big(\tau=\dfrac{-1+\sqrt{11}\,i}{6}\Big)&=-\frac{\sqrt{11}-5 i}{6}\\
&\qquad+\frac{(2\sqrt{3}+\sqrt{11}+i)(3 \sqrt{33}-17)^{1/3}}{3}\\
&\qquad\quad+\frac{(2\sqrt{3}-\sqrt{11}-i)(17+3 \sqrt{33}\>\!\>\!)^{1/3}}{3}
\\
\\
k(\tau=2\sqrt{11}i)&=\frac{\varepsilon_1+\varepsilon_2(3\sqrt{33}-17)^{1/3}-\varepsilon_3(17+3\sqrt{33})^{1/3}}{3}\\
k\Big(\tau=\frac{\sqrt{11}}{\,\,22}i\Big)&=\frac{(\sqrt{22}-3\sqrt{2})^{1/2}}{3}\Big(\varepsilon_4-\varepsilon_5(3\sqrt{33}-17)^{1/3}\\
&\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon_6(17+3\sqrt{33})^{1/3}\Big)\\
\\
\varepsilon_1&=2093+632 \sqrt{11}-1482 \sqrt{2}-446 \sqrt{22}\\
\varepsilon_2&=1376+1168 \sqrt{3}+416 \sqrt{11}+352 \sqrt{33}\\
&\qquad\>\>-975 \sqrt{2}-825 \sqrt{6}-293 \sqrt{22}-249 \sqrt{66}\\
\varepsilon_3&=1376+825 \sqrt{6}+416 \sqrt{11}+249 \sqrt{66}\\
&\qquad\quad-975 \sqrt{2}-1168 \sqrt{3}-293 \sqrt{22}-352 \sqrt{33}\\
\\
\varepsilon_{4}&=3132+944 \sqrt{11}-2211 \sqrt{2}-667 \sqrt{22}\\
\varepsilon_5&=1455 \sqrt{2}+1234 \sqrt{6}+439 \sqrt{22}+372 \sqrt{66}\\
&\qquad\>\>-2058-1744 \sqrt{3}-620 \sqrt{11}-526 \sqrt{33}\\
\varepsilon_6&=2058+1234 \sqrt{6}+620 \sqrt{11}+372 \sqrt{66}\\
&\qquad-1455 \sqrt{2}-1744 \sqrt{3}-439 \sqrt{22}-526 \sqrt{33}\\

\end{align*}


\begin{align*}
j
\big(2\sqrt{2}\,i\big)
&=10^3 (19+13 \sqrt{2}\>\!)^3\\
j
\big(3\,i\big)
&=2^4(1+\sqrt{3}\>\!)^4(21+20 \sqrt{3}\>\!\>\!)^3\\
j
\big(\sqrt{10}\,i\big)
&=6^3 (65 + 27\sqrt{5}\>\!)^3\\
j\big(\sqrt{11}\,i\big)&=\left(\frac{1024+(697+177\sqrt{33}\>\!) (3\sqrt{33}-17\>\!\>\!)^{1/3}}{3}\right.\\
&\qquad\qquad\left.+\dfrac{(177\sqrt{33}-697)(17+3\sqrt{33}\>\!\>\!)^{1/3}}{3}\right)^3\\
j
\big(2\sqrt{3}\,i\big)
&=2^2\cdot15^3(30+17\sqrt{3}\>\!)^3\\
j
\big(\sqrt{13}\,i\big)
&=30^3(31+9 \sqrt{13}\>\!)^3\\
\\
j
\bigg(\frac{1+\sqrt{11}\,i}{2}\bigg)
&=-32^3\\
j
\big(2\sqrt{11}\,i\big)

&=\left(\frac{52(10388+3131 \sqrt{11})}{3}\right.\\
&\qquad\quad\left.+\frac{\rho_1(3 \sqrt{33}-17)^{1/3}}{6}
+\frac{\rho_2(17+3 \sqrt{33})^{1/3}}{6}\right)^3\\
\rho_1&=(710362+602351 \sqrt{3}+214267 \sqrt{11}+181574 \sqrt{33}\,)\\
\rho_2&=(602351 \sqrt{3}+181574 \sqrt{33}-710362-214267 \sqrt{11})\\
\end{align*}

1. Sqrt[ModularLambda[(1 + I Sqrt[11])/2]] // N
2. Sqrt[1/2 (1 - Sqrt[
3.     1/3 (-521 -
4.        4 (1 - I Sqrt[3]) (43 - 11 Sqrt[33]) (3 Sqrt[33] + 17)^(1/3) +
5.        4 (1 + I Sqrt[3]) (43 + 11 Sqrt[33]) (3 Sqrt[33] - 17)^(
6.         1/3))])] // N
7. Sqrt[2 (2 - 4 x + 4 x^2 - x^3)] /. {x ->
8.     2/3 - (2 (3 Sqrt[33] + 13))^(1/3)/6 + (2 (3 Sqrt[33] - 13))^(1/3)/
9.      6} // FullSimplify
10. (2 Sqrt[11] - (Sqrt[11] + Sqrt[3]) (3 Sqrt[33] - 17)^(
11.    1/3) + (Sqrt[11] - Sqrt[3]) (17 + 3 Sqrt[33])^(
12.    1/3))/6 // FullSimplify
13. (5 - 2 x)/11 Sqrt[1 - 4 x + 4 x^2 - 2 x^3] + (3 - 2 x)/11 Sqrt[
14.     2 (2 - 4 x + 4 x^2 - x^3)] /. {x ->
15.     2/3 - (2 (3 Sqrt[33] + 13))^(1/3)/6 + (2 (3 Sqrt[33] - 13))^(1/3)/
16.      6} // FullSimplify
17. N[(1 - k^2) (1 - ((1 - Sqrt[1 - s^2])/(1 + Sqrt[1 - s^2]))^2) - (1 -
18.      k^(1/2) ((1 - Sqrt[1 - s^2])/(1 + Sqrt[1 - s^2]))^(
19.       1/2))^4 /. {k -> Sqrt[ModularLambda[1 + Sqrt[11] I]],
20.    s -> Sqrt[ModularLambda[(1 + Sqrt[11] I)/6]]}, 20]
21. N[Sqrt[ModularLambda[1 + Sqrt[11] I]], 20]
22. N[(-((7 Sqrt[11] - 22)/3) + (
23.     4 (4 + Sqrt[33] - 3 Sqrt[3] - Sqrt[11]) (3 Sqrt[33] - 17)^(1/3))/
24.     3 - (4 (4 + 3 Sqrt[3] - Sqrt[11] - Sqrt[33]) (17 + 3 Sqrt[33])^(
25.      1/3))/3) I, 20]
26. N[Sqrt[ModularLambda[(1 + Sqrt[11] I)/6]], 20]
27. N[-((Sqrt[11] - 5 I)/
28.    6) + ((2 Sqrt[3] + Sqrt[11] + I) (3 Sqrt[33] - 17)^(1/3))/
29.   3 + ((2 Sqrt[3] - Sqrt[11] - I) (17 + 3 Sqrt[33])^(1/3))/3, 20]

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1. N[(w1 + w2 (3 Sqrt[33] - 17)^(1/3) - w3 (3 Sqrt[33] + 17)^(1/3))/3 /. {
2.    w1 -> 2093 + 632 Sqrt[11] - 1482 Sqrt[2] - 446 Sqrt[22],
3.    w2 ->
4.     1376 + 1168 Sqrt[3] + 416 Sqrt[11] + 352 Sqrt[33] - 975 Sqrt[2] -
5.      825 Sqrt[6] - 293 Sqrt[22] - 249 Sqrt[66],
6.    w3 ->
7.     1376 + 825 Sqrt[6] + 416 Sqrt[11] + 249 Sqrt[66] - 975 Sqrt[2] -
8.      1168 Sqrt[3] - 293 Sqrt[22] - 352 Sqrt[33]}, 50]
9. N[Sqrt[ModularLambda[2 Sqrt[11] I]], 50]
10. N[(Sqrt[22] - 3 Sqrt[2])^(1/2) (
11.    w4 - w5 (3 Sqrt[33] - 17)^(1/3) - w6 (3 Sqrt[33] + 17)^(1/3))/3 /. {
12.    w4 -> 3132 + 944 Sqrt[11] - 2211 Sqrt[2] - 667 Sqrt[22],
13.    w5 ->
14.     1455 Sqrt[2] + 1234 Sqrt[6] + 439 Sqrt[22] + 372 Sqrt[66] -
15.      2058 - 1744 Sqrt[3] - 620 Sqrt[11] - 526 Sqrt[33],
16.    w6 ->
17.     2058 + 1234 Sqrt[6] + 620 Sqrt[11] + 372 Sqrt[66] -
18.      1455 Sqrt[2] - 1744 Sqrt[3] - 439 Sqrt[22] - 526 Sqrt[33]}, 50]
19. N[Sqrt[ModularLambda[Sqrt[11]/22 I]], 50]

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青青子衿

\begin{align*}
j(\tau)&=\frac{\left(\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^{12}+250 \left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^6+3125\right)^3}{\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^{30}}\\
j(5\tau)&=\frac{\left(\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^{12}+10 \left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^6+5\right)^3}{\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^6}
\end{align*}

\begin{align*}
0&=u^6 - v^6 + 5 u^2 v^2 (u^2 - v^2) + 4 u v (1 - u^4 v^4)\\
\\
u&=\left(\frac{1}{2}+\frac{1-11s-s^{2}}{2(1+2s)^{2}}\sqrt{\frac{1+s^{2}}{1+2s}}\right)^{\frac{1}{8}}\\
v&=
\left(\frac{1}{2}+\frac{1+s-s^{2}}{2}\sqrt{\frac{1+s^{2}}{1+2s}}\right)^{\frac{1}{8}}
\end{align*}

1. (((DedekindEta[0.1 I]/DedekindEta[0.5 I])^6)^2 +
2.   250*(DedekindEta[0.1 I]/DedekindEta[0.5 I])^6 +
3.   3125)^3/((DedekindEta[0.1 I]/DedekindEta[0.5 I])^6)^5
4. 1728 KleinInvariantJ[0.1 I]
5. (((DedekindEta[0.1 I]/DedekindEta[0.5 I])^6)^2 +
6.   10*(DedekindEta[0.1 I]/DedekindEta[0.5 I])^6 + 5)^3/(DedekindEta[
7.   0.1 I]/DedekindEta[0.5 I])^6
8. 1728 KleinInvariantJ[0.5 I]
9. u^6-v^6+5*u^2*v^2*(u^2-v^2)+4*u*v*(1-u^4*v^4)=0
10. x^6-y^6+5*x^2*y^2*(x^2-y^2)+4*x*y*(1-x^4*y^4)=0
11. u^4*v^4+((1-u^8)*(1-v^8))^{1/2}+2*(16*u^8*v^8*(1-u^8)*(1-v^8))^{1/6}=1
12. x^4*y^4+((1-x^8)*(1-y^8))^{1/2}+2*(16*x^8*y^8*(1-x^8)*(1-y^8))^{1/6}=1
13. N[Normal[Series[(12*(1 + 240*Sum[(k^3*q^k)/(1 - q^k), {k, 1, 100}]))/
14.            ((1 + 240*Sum[(k^3*q^k)/(1 - q^k), {k, 1, 100}])^3 - (1 -
15.            504*Sum[(k^5*q^k)/(1 - q^k), {k, 1, 100}])^2)^(1/3),
16.          {q, 0, 20}]] /. q -> Exp[2*Pi*I*(I/2)]]

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青青子衿

青青子衿 发表于 2023-9-11 10:26
\begin{align*}
k\big(\tau=\sqrt{11}\,i\big)&=\frac{3\sqrt{2}+\sqrt{22}}{12}\\
&\qquad-\frac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}\\
&\qquad\qquad-\frac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\\
k\Big(\tau=\frac{\sqrt{11}}{\,\,11}i\Big)&=\frac{\sqrt{22}-3 \sqrt{2}}{12}\\
&\qquad+\frac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}\\
&\qquad\qquad+\frac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\\
\end{align*}

\begin{gather*}

\begin{split}
u^{2}v^{2}&=\frac{1-2 \chi}{2}-\frac{\sqrt{\small1-4\chi+4\chi^{2}-2\chi^{3}}}{2}\\
u^4v^4&=\frac{(1-\chi ) \left(1-3 \chi +\chi ^2\right)}{2} -\frac{(1-2 \chi )\sqrt{\small1-4\chi+4\chi^{2}-2\chi^{3}}}{2}\\

u^{8}v^{8}&=\frac{2-16\chi+48\chi^{2}-68\chi^{3}+48\chi^{4}-16\chi^{5}+\chi^{6}}{4}\\
&\qquad\quad-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}\\
u^{8}+v^{8}&=1-(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}\\
\\
u^{8}&=\frac{1}{2}-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}\\
&\qquad-\frac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}\\
v^{8}&=\frac{1}{2}-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}\\
&\qquad+\frac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}\\
\end{split}\\
\\
\\
\begin{split}
\left(u^{8}v^{8}\right)^{1/4}+\left(\left(1-u^{8}\right)\left(1-v^{8}\right)\right)^{1/4}+2\left(16u^{8}v^{8}\left(1-u^{8}\right)\left(1-v^{8}\right)\right)^{1/12}=1
\end{split}
\end{gather*}


\begin{align*}
&\left(v^{4}-u^{4}+2uv(1-u^{2}v^{2})\right)^{3}\\
&\qquad+16uv(1-u^{2}v^{2})\left((v^{4}-u^{4})^{2}-2(1+u^{2}v^{2})^{4}\right)\\
&\qquad\qquad\quad+32u^{2}v^{2}(v^{4}-u^{4})\left(8u^{2}v^{2}+(1-u^{2}v^{2})^{2}\right)=0
\end{align*}

1. k_{a}=\frac{3\sqrt{2}+\sqrt{22}}{12}-\frac{(\sqrt{2}+\sqrt{6})(\sqrt{3}+\sqrt{11})(3\sqrt{33}-17)^{1/3}}{24}-\frac{(\sqrt{6}-\sqrt{2})(\sqrt{11}-\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
2. k_{b}=\frac{\sqrt{22}-3\sqrt{2}}{12}+\frac{(\sqrt{6}-\sqrt{2})(\sqrt{3}+\sqrt{11})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(\sqrt{2}+\sqrt{6})(\sqrt{11}-\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
3. \left(k_{a}^{1/4},k_{b}^{1/4}\right)
4. \left(x^{8}y^{8}\right)^{\frac{1}{4}}+\left(\left(1-x^{8}\right)\left(1-y^{8}\right)\right)^{\frac{1}{4}}+2\left(16x^{8}y^{8}\left(1-x^{8}\right)\left(1-y^{8}\right)\right)^{\frac{1}{12}}=1
5. \chi=0.34
6. u=\left(\frac{1}{2}-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}-\frac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}\right)^{\frac{1}{8}}
7. v=\left(\frac{1}{2}-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}+\frac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}\right)^{\frac{1}{8}}
8. \left(u^{8}v^{8}\right)^{\frac{1}{4}}+\left(\left(1-u^{8}\right)\left(1-v^{8}\right)\right)^{\frac{1}{4}}+2\left(16u^{8}v^{8}\left(1-u^{8}\right)\left(1-v^{8}\right)\right)^{\frac{1}{12}}-1

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\begin{align*}
d_1 &= 2^{1/3}\left( \frac{\alpha ^{11}(1-\beta )^{11}}{\beta (1-\alpha )}\right) ^{1/24}\\
&=2-3a+2a^2+\frac{2-a}{2}\sqrt{\small2(2-4a+4a^2-a^3)}\\
d_2 &= 2^{1/3} \left( \frac{\beta ^{11}(1-\alpha )^{11}}{\alpha (1-\beta )}\right) ^{1/24}\\
&=2-3a+2a^2-\frac{2-a}{2}\sqrt{\small2(2-4a+4a^2-a^3)}\\
\frac{1}{M}&=-(5-2\chi)\sqrt{\small1-4\chi+4\chi^{2}-2\chi^{3}}\\
&\qquad\qquad+(3-2\chi)\sqrt{\small2(2-4\chi+4\chi^{2}-\chi^{3})}\\
M&=\frac{5-2\chi}{11}\sqrt{\small1-4\chi+4\chi^{2}-2\chi^{3}}\\
&\qquad\qquad+\frac{3-2\chi}{11}\sqrt{\small2(2-4\chi+4\chi^{2}-\chi^{3})}\\

\end{align*}

1. \chi=0.3
2. \alpha_{1}=\frac{(3-2\chi)\delta_{2}}{2}-\frac{(5-2\chi)\delta_{1}}{2}-\frac{1}{2}
3. \alpha_{2}=(1-\chi)^{2}(3-2\chi)\delta_{2}-(1-\chi)^{2}(5-2\chi)\delta_{1}-1
4. \alpha_{3}=\varepsilon_{30}+\varepsilon_{31}\delta_{1}-\varepsilon_{32}\delta_{2}-\varepsilon_{33}\delta_{1}\delta_{2}
5. \alpha_{4}=\varepsilon_{40}+\varepsilon_{41}\delta_{1}-\varepsilon_{42}\delta_{2}-\varepsilon_{43}\delta_{1}\delta_{2}
6. \alpha_{5}=\varepsilon_{52}\delta_{2}-\varepsilon_{50}-\varepsilon_{51}\delta_{1}+\varepsilon_{53}\delta_{1}\delta_{2}
7. \delta_{1}=\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}
8. \delta_{2}=\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}
9. \varepsilon_{30}=\frac{22-98\chi+205\chi^{2}-234\chi^{3}+156\chi^{4}-56\chi^{5}+8\chi^{6}}{2}
10. \varepsilon_{31}=\frac{(5-2\chi)(2-2\chi+\chi^{2})}{2}
11. \varepsilon_{32}=\frac{(3-2\chi)(2-2\chi+\chi^{2})}{2}
12. \varepsilon_{33}=(5-10\chi+8\chi^{2}-2\chi^{3})
13. \varepsilon_{40}=\frac{11-66\chi+187\chi^{2}-305\chi^{3}+302\chi^{4}-182\chi^{5}+60\chi^{6}-8\chi^{7}}{2}
14. \varepsilon_{41}=\frac{(1-\chi)^{2}(5-2\chi)}{2}
15. \varepsilon_{42}=\frac{(1-\chi)^{2}(3-2\chi)}{2}
16. \varepsilon_{43}=\frac{(1-\chi)(5-13\chi+14\chi^{2}-4\chi^{3})}{2}
17. \varepsilon_{50}=\frac{(2-\chi)(16-74\chi+159\chi^{2}-190\chi^{3}+134\chi^{4}-52\chi^{5}+8\chi^{6})}{4}
18. \varepsilon_{51}=\frac{(2-\chi)(16-54\chi+93\chi^{2}-88\chi^{3}+44\chi^{4}-8\chi^{5})}{4}
19. \varepsilon_{52}=\frac{17-76\chi+161\chi^{2}-193\chi^{3}+136\chi^{4}-52\chi^{5}+8\chi^{6}}{4}
20. \varepsilon_{53}=\frac{15-38\chi+43\chi^{2}-22\chi^{3}+4\chi^{4}}{4}
21. y_{11}\left(x,\varphi\right)=\frac{1-\frac{1-x}{1+x}\left(\frac{1-\alpha_{1}x+\alpha_{2}x^{2}-\alpha_{3}x^{3}+\alpha_{4}x^{4}-\alpha_{5}x^{5}}{1+\alpha_{1}x+\alpha_{2}x^{2}+\alpha_{3}x^{3}+\alpha_{4}x^{4}+\alpha_{5}x^{5}}\right)^{2}}{1+\frac{1-x}{1+x}\left(\frac{1-\alpha_{1}x+\alpha_{2}x^{2}-\alpha_{3}x^{3}+\alpha_{4}x^{4}-\alpha_{5}x^{5}}{1+\alpha_{1}x+\alpha_{2}x^{2}+\alpha_{3}x^{3}+\alpha_{4}x^{4}+\alpha_{5}x^{5}}\right)^{2}}
22. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
23. M\int_{0}^{y_{11}\left(x,1\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}dt
24. M=\frac{(3-2\chi)\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}+(5-2\chi)\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}}{11}
25. u=\left(\frac{1}{2}-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}-\frac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}\right)^{\frac{1}{8}}
26. v=\left(\frac{1}{2}-\frac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\sqrt{1-4\chi+4\chi^{2}-2\chi^{3}}+\frac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\sqrt{2(2-4\chi+4\chi^{2}-\chi^{3})}\right)^{\frac{1}{8}}

复制代码

\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}
=M\int_{0}^{y_{11}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Vt^{2}\right)}}\\
\\

\frac{1-y_{11}}{1+y_{11}}=\frac{1-x}{1+x}\left(\frac{1-\alpha_1x+\alpha_2x^{2}-\alpha_3x^{3}+\alpha_4x^{4}-\alpha_5x^{5}}{1+\alpha_1x+\alpha_2x^{2}+\alpha_3x^{3}+\alpha_4x^{4}+\alpha_5x^{5}}\right)^2\\
\\
\\
\qquad\begin{split}
\delta_1&=\sqrt{\scriptsize1-4\chi+4\chi^{2}-2\chi^{3}}\\
\delta_2&=\sqrt{\scriptsize2(2-4\chi+4\chi^{2}-\chi^{3})}\\
M&=\tfrac{5-2\chi}{11}\delta_1+\tfrac{3-2\chi}{11}\delta_2\\
U&=\tfrac{1}{2}-\tfrac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\delta_1\\
&\qquad\,-\tfrac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\delta_2\\
V&=\tfrac{1}{2}-\tfrac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})}{2}\delta_1\\
&\qquad\,+\tfrac{\chi(2-\chi)(2-3\chi+2\chi^{2})}{2}\delta_2\\
\alpha_1&=-\tfrac{5-2 \chi}{2}\delta_1+\tfrac{3-2 \chi}{2}\delta_2-\tfrac{1}{2}\\
\alpha_2&=-{\scriptsize(1-\chi)^2(5-2 \chi )}\delta_1+{\scriptsize(1-\chi )^2(3-2 \chi ) }\delta_2 -{\scriptsize1}\\
\alpha_3&=\tfrac{22-98 \chi+205 \chi ^2-234 \chi ^3+156 \chi ^4 -56 \chi ^5+8 \chi ^6}{2}\\
&\qquad+\tfrac{(5-2 \chi )(2-2 \chi +\chi ^2)}{2}\delta_1\\
&\qquad\quad-\tfrac{(3-2 \chi )(2-2 \chi +\chi ^2)}{2}\delta_2\\
&\qquad\qquad-\left({\scriptsize5-10\chi+8\chi^2-2 \chi ^3}\right) \delta_1\delta_2\\

\alpha_4&=\tfrac{11-66 \chi+187 \chi ^2-305 \chi ^3+302 \chi ^4-182 \chi ^5+60 \chi ^6-8 \chi ^7}{2}\\
&\qquad+\tfrac{(1-\chi)^2(5-2 \chi )}{2}\delta_1\\
&\qquad\quad-\tfrac{(1-\chi)^2(3-2 \chi)}{2}\delta_2\\
&\qquad\qquad-\tfrac{(1-\chi )(5-13 \chi+14 \chi ^2-4 \chi ^3)}{2}\delta_1\delta_2\\

\alpha_5&=-\tfrac{(2-\chi )(16-74 \chi+159 \chi ^2-190 \chi ^3+134 \chi ^4-52 \chi ^5+8 \chi ^6)}{4}\\
&\qquad-\tfrac{(2-\chi )(16-54 \chi+93 \chi ^2-88 \chi ^3+44 \chi ^4 -8 \chi ^5)}{4} \delta_1\\
&\qquad\quad+\tfrac{17-76 \chi+161 \chi ^2-193 \chi ^3+136 \chi ^4-52 \chi ^5+8 \chi ^6}{4}\delta_2\\
&\qquad\qquad+\tfrac{15-38 \chi+43 \chi ^2-22 \chi ^3+4 \chi ^4}{4}\delta_1\delta_2\\
\end{split}

\end{gather*}


\begin{align*}

\begin{split}
\omega_1&=\alpha _1^2+2 \alpha _1 \alpha _2+2 \alpha _2+2 \alpha _3\\
\omega_2&=\alpha _2^2+2 \alpha _1 \alpha _3+2 \alpha _1 \alpha _4+2 \alpha _2 \alpha _3+2 \alpha _4+2 \alpha _5\\
\omega_3&=\alpha _3^2+2 \alpha _1 \alpha _5+2 \alpha _2 \alpha _4+2 \alpha _2 \alpha _5+2 \alpha _3 \alpha _4\\
\omega_4&=\alpha _4^2+2 \alpha _3 \alpha _5+2 \alpha _4 \alpha _5\\
\omega_5&=\alpha _5^2\\
\tau_1&=\alpha _1^2+2 \alpha _1+2 \alpha _2\\
\tau_2&=\alpha _2^2+2 \alpha _1 \alpha _2+2 \alpha _1 \alpha _3+2 \alpha _3+2 \alpha _4\\
\tau_3&=\alpha _3^2+2 \alpha _1 \alpha _4+2 \alpha _1 \alpha _5+2 \alpha _2 \alpha _3+2 \alpha _2 \alpha _4+2 \alpha _5\\
\tau_4&=\alpha _4^2+2 \alpha _2 \alpha _5+2 \alpha _3 \alpha _5+2 \alpha _3 \alpha _4\\
\tau_5&=\alpha _5^2+2 \alpha _4 \alpha _5\\
\end{split}\\
\end{align*}


\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}
=M\int_{0}^{y_{11}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Vt^{2}\right)}}\\
\\
y_{11}=\frac{x(\frac{1}{M}+\omega_1x^2+\omega_2x^4+\omega_3x^6+\omega_4x^8+\omega_5x^{10})}{1+\tau_1x^2+\tau_2x^4+\tau_3x^6+\tau_4x^8+\tau_5x^{10}}\\
\\
\\
\begin{split}
\delta_1&=\sqrt{\scriptsize1-4\chi+4\chi^{2}-2\chi^{3}}\\
\delta_2&=\sqrt{\scriptsize2(2-4\chi+4\chi^{2}-\chi^{3})}\\

M&=\tfrac{(5-2\chi)\delta_1}{11}+\tfrac{(3-2\chi)\delta_2}{11}\\
\tfrac{1}{M}&=-{\scriptsize\,(5-2\chi)\delta_1}+{\scriptsize(3-2\chi)\delta_2}\\
U&=\tfrac{1}{2}-\tfrac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})\delta_1}{2}\\
&\qquad\>\>-\tfrac{\chi(2-\chi)(2-3\chi+2\chi^{2})\delta_2}{2}\\
V&=\tfrac{1}{2}-\tfrac{(1-\chi)(1-2\chi)(1-3\chi+\chi^{2})\delta_1}{2}\\
&\qquad\>\>+\tfrac{\chi(2-\chi)(2-3\chi+2\chi^{2})\delta_2}{2}\\
\end{split}\\
\\
\qquad\qquad\begin{split}

\omega_1&=\tfrac{1}{2}{\scriptsize(195-1040 \chi+2412 \chi ^2-3150 \chi ^3)}\\
&\qquad+\tfrac{1}{2}{\scriptsize(2496 \chi ^4-1184 \chi ^5+304 \chi ^6-32 \chi ^7)}\\
&\qquad\>\>+\tfrac{5}{2}{\scriptsize(5-2 \chi )\delta_1}-\tfrac{5}{2}{\scriptsize(3-2 \chi )\delta_2}\\
&\qquad\quad-\tfrac{1}{2}{\scriptsize(16 \chi ^4-104 \chi ^3+240 \chi ^2-240 \chi +95)\delta_1\delta_2}\\

\omega_2&=-{\scriptsize\,(195-1040\chi+2412 \chi ^2-3150 \chi ^3)}\\
&\qquad-{\scriptsize\,(2496 \chi ^4-1184\chi^5+304 \chi ^6-32 \chi ^7)}\\
&\qquad\>\>-{\scriptsize\,(421-2518\chi+7292\chi^{2}-12777\chi^{3}+14725\chi^{4})\delta_1}\\
&\qquad\quad+{\scriptsize\,(11472\chi^{5}-5988\chi^{6}+2000\chi^{7}-384\chi^{8}+32\chi^{9})\delta_1}\\
&\qquad\quad\>\>+{\scriptsize\,(1-\chi )(213-1249\chi+3355\chi^{2})\delta_2}\\
&\qquad\quad-{\scriptsize\,(1-\chi )(5239\chi^{3}-5212\chi^{4}+3372\chi^{5})\delta_2}\\
&\qquad\>\>+{\scriptsize\,(1-\chi )(1376\chi^{6}-320\chi^{7}+32\chi^{8})\delta_2}\\
&\qquad+{\scriptsize\,(95-240 \chi+240 \chi ^2-104 \chi ^3+16 \chi ^4)\delta_1\delta_2}\\

\omega_3&=\tfrac{1}{2}{\scriptsize(1578-14620 \chi+65644 \chi ^2-187895 \chi ^3+378995 \chi ^4)}\\
&\quad\>\>\,-\tfrac{1}{2}{\scriptsize(564302 \chi ^5-634168 \chi ^6+542090 \chi ^7-351060 \chi ^8)}\\
&\qquad-\tfrac{1}{2}{\scriptsize(169440 \chi ^9-58960 \chi ^{10}+13920 \chi ^{11}-1984 \chi ^{12}+128 \chi ^{13})}\\
&\qquad\quad+\tfrac{1}{2}\scriptsize{(1238-7544 \chi+21876 \chi ^2-38331 \chi ^3+44175 \chi ^4)\delta_1}\\
&\qquad\quad\>\>-\tfrac{1}{2}{\scriptsize(34416 \chi ^5-17964 \chi ^6+6000 \chi ^7-1152 \chi ^8+96 \chi ^9)\delta_1}\\
&\qquad\qquad-\tfrac{1}{2}{\scriptsize(624-4376 \chi+13812 \chi ^2-25782 \chi ^3+31353 \chi ^4)\delta_2}\\
&\qquad\quad\>\>\>+\tfrac{1}{2}{\scriptsize(25752 \chi ^5-14244 \chi ^6+5088 \chi ^7-1056 \chi ^8+96 \chi ^9) \delta_2}\\
&\qquad\quad\>-\tfrac{1}{2}{\scriptsize(784-4968 \chi+15940 \chi ^2-32370 \chi ^3)\delta_1\delta_2}\\
&\qquad\>\>\>-\tfrac{1}{2}{\scriptsize(44849 \chi ^4-43622 \chi ^5+29904 \chi ^6)\delta_1\delta_2}\\
&\quad\quad+\tfrac{1}{2}{\scriptsize(14168 \chi ^7-4400 \chi ^8+800 \chi ^9-64 \chi ^{10})\delta_1\delta_2}\\

\omega_4&=-\tfrac{1}{2}{\scriptsize(1383 - 13580\chi + 63232\chi^2 - 184745\chi^3 + 376499\chi^4)}\\
&\quad\>\>\>+\tfrac{1}{2}{\scriptsize(563118 \chi ^5-633864 \chi ^6+542058 \chi ^7-351060 \chi ^8)}\\
&\qquad\>+\tfrac{1}{2}{\scriptsize(169440 \chi ^9-58960 \chi ^{10}+13920 \chi ^{11}-1984 \chi ^{12}+128 \chi ^{13})}\\
&\qquad\quad-\tfrac{1}{4}\scriptsize{(2866-23428 \chi+95608 \chi ^2-253068 \chi ^3+480002 \chi ^4)\delta_1}\\
&\qquad\quad\>\>+\tfrac{1}{4}{\scriptsize(682920 \chi ^5-743349 \chi ^6+621412 \chi ^7-395604 \chi ^8)\delta_1}\\
&\qquad\qquad\>+\tfrac{1}{4}{\scriptsize(187848 \chi ^9-64128 \chi ^{10}+14784 \chi ^{11}-2048 \chi ^{12}+128 \chi ^{13})\delta_1}\\
&\qquad\qquad\>\>\>+\tfrac{1}{4}{\scriptsize(1438-13132 \chi +57400 \chi ^2-159930 \chi ^3+316474 \chi ^4)\delta_2}\\
&\qquad\qquad\>-\tfrac{1}{4}{\scriptsize(467648 \chi ^5-527705 \chi ^6+457424 \chi ^7-302588 \chi ^8) \delta_2}\\
&\qquad\quad\>\>\>-\tfrac{1}{4}{\scriptsize(149880 \chi ^9-53664 \chi ^{10}+13056 \chi ^{11}-1920 \chi ^{12}+128 \chi ^{13}) \delta_2}\\
&\qquad\quad\>+\tfrac{1}{2}{\scriptsize(689-4728 \chi+15700 \chi ^2-32266 \chi ^3)\delta_1\delta_2}\\
&\qquad\>\>\>+\tfrac{1}{2}{\scriptsize(44833 \chi ^4-43622 \chi ^5+29904 \chi ^6)\delta_1\delta_2}\\
&\quad\quad-\tfrac{1}{2}{\scriptsize(14168 \chi ^7-4400 \chi ^8+800 \chi ^9-64 \chi ^{10})\delta_1\delta_2}\\

\omega_5&=\tfrac{1-2 \chi }{8} {\scriptsize(2052-18464 \chi+81792 \chi ^2-233290 \chi ^3+474904 \chi ^4)}\\
&\quad\>\>\>-\tfrac{1-2 \chi }{8}{\scriptsize(725800 \chi ^5-854935 \chi ^6+785260 \chi ^7-562876 \chi ^8)}\\
&\qquad\>-\tfrac{1-2 \chi }{8}{\scriptsize(311994 \chi ^9-131008 \chi ^{10}+40192 \chi ^{11})}\\
&\qquad\quad+\tfrac{1-2 \chi }{8}\scriptsize{(8464 \chi ^{12}-1088 \chi ^{13}+64 \chi ^{14})}\\
&\qquad\quad\>\>\>+\tfrac{1}{8}{\scriptsize(2044-18400 \chi+81024 \chi ^2 -227514 \chi ^3)\delta_1}\\
&\qquad\qquad\>+\tfrac{1}{8}{\scriptsize(450552 \chi ^4-659976 \chi ^5+731373 \chi ^6)\delta_1}\\
&\qquad\qquad\>\>\>-\tfrac{1}{8}{\scriptsize(617412 \chi ^7-394836 \chi ^8+187784 \chi ^9)\delta_1}\\
&\qquad\qquad\>+\tfrac{1}{8}{\scriptsize(64128 \chi ^{10}-14784 \chi ^{11}+2048 \chi ^{12}-128 \chi ^{13}) \delta_1}\\

&\qquad\quad\>\>\>-\tfrac{(2-\chi ) (1-2 \chi )(2-3 \chi +2 \chi ^2)}{8}{\scriptsize(256-1530\chi+4456 \chi ^2) \delta_2}\\
&\qquad\quad\>+\tfrac{(2-\chi ) (1-2 \chi )(2-3 \chi +2 \chi ^2)}{8}{\scriptsize(8040 \chi ^3-9735 \chi ^4+8096 \chi ^5)\delta_2}\\
&\qquad\>\>\>-\tfrac{(2-\chi ) (1-2 \chi )(2-3 \chi +2 \chi ^2)}{8}{\scriptsize(4576 \chi ^6-1672 \chi ^7+352 \chi ^8-32 \chi ^9)\delta_2}\\
&\quad\quad-\tfrac{(2-\chi ) (2-3 \chi +2 \chi ^2)}{8}{\scriptsize(256-1542 \chi+4504 \chi ^2-8088 \chi ^3)\delta_1\delta_2}\\
&\quad\quad\>\>\>-\tfrac{(2-\chi ) (2-3 \chi +2 \chi ^2)}{8}{\scriptsize(9747 \chi ^4-8096 \chi ^5+4576 \chi ^6)\delta_1\delta_2}\\
&\quad\qquad\>+\tfrac{(2-\chi ) (2-3 \chi +2 \chi ^2)}{8}{\scriptsize(1672 \chi ^7-352 \chi ^8+32 \chi ^9)\delta_1\delta_2}\\

\tau_1&=\tfrac{1}{2}{\scriptsize(5-2 \chi )^2(1-4 \chi+4 \chi ^2 -2 \chi ^3)}-\tfrac{1}{2}{\scriptsize(5-2 \chi )(5-8 \chi +4 \chi ^2)\delta_1}\\
&\quad\>\>+\tfrac{1}{2}{\scriptsize(3-2 \chi )(5-8 \chi +4 \chi ^2)\delta_2}-\tfrac{1}{2}{\scriptsize(3-2 \chi )(5-2 \chi )\delta_1\delta_2}\\

\tau_2&={\scriptsize85-660 \chi+2254 \chi ^2-4358 \chi ^3+5365 \chi ^4}\\
&\quad\>\>\>-{\scriptsize\,4396 \chi ^5+2408 \chi ^6-852 \chi ^7+176 \chi ^8-16 \chi ^9}\\
&\qquad\>-{\scriptsize(85-490 \chi+1186 \chi ^2-1571 \chi ^3)\delta_1}\\
&\qquad\quad-{\scriptsize(1248 \chi ^4-592 \chi ^5+152 \chi ^6-16 \chi ^7)\delta_1}\\
&\qquad\quad\>\>+{\scriptsize(40-290 \chi+774 \chi ^2-1103 \chi ^3)\delta_2}\\
&\qquad\qquad+{\scriptsize(936 \chi ^4-480 \chi ^5+136 \chi ^6-16 \chi ^7)\delta_2}\\
&\qquad\qquad\>\>-{\scriptsize2(20-105 \chi+199 \chi ^2-194 \chi ^3)\delta_1\delta_2}\\
&\qquad\qquad\quad-{\scriptsize2(107 \chi ^4-32 \chi ^5+4\chi ^6)\delta_1\delta_2}\\

\tau_3&=\tfrac{1}{2}{\scriptsize(226-2772 \chi+14556 \chi ^2-44885 \chi ^3+91241 \chi ^4)}\\
&\qquad-\tfrac{1}{2}{\scriptsize(129888 \chi ^5-133658 \chi ^6+100650 \chi ^7-55216 \chi ^8)}\\
&\qquad\>\>\>-\tfrac{1}{2}{\scriptsize(21544 \chi ^9-5664 \chi ^{10}+896 \chi ^{11}-64 \chi ^{12})}\\
&\qquad\quad\>-\tfrac{1}{2}{\scriptsize(226-2320 \chi+9916 \chi ^2-24211 \chi ^3)\delta_1}\\
&\qquad\quad\>\>\>-\tfrac{1}{2}{\scriptsize(37783 \chi ^4-39738 \chi ^5+28628 \chi ^6)\delta_1}\\
&\qquad\qquad\>+\tfrac{1}{2}{\scriptsize(13944 \chi ^7-4384 \chi ^8+800 \chi ^9-64 \chi ^{10})\delta_1}\\
&\qquad\qquad\>\>\>+\tfrac{1}{2}{\scriptsize(118-1268 \chi +5948 \chi ^2-15588 \chi ^3)\delta_2}\\
&\qquad\qquad\quad+\tfrac{1}{2}{\scriptsize(25767\chi^{4}-28526\chi^{5}+21644\chi^{6})\delta_2}\\
&\qquad\qquad\>\>\,-\tfrac{1}{2}{\scriptsize(11160\chi^{7}-3744\chi^{8}+736\chi^{9}-64\chi^{10})\delta_2}\\
&\qquad\qquad-\tfrac{1}{2}{\scriptsize(118-1032 \chi+3884 \chi ^2-8010 \chi ^3+10227 \chi ^4)\delta_1\delta_2}\\
&\qquad\quad\>\>+\tfrac{1}{2}{\scriptsize(8552 \chi ^5-4748 \chi ^6+1696 \chi ^7-352 \chi ^8+32 \chi ^9)\delta_1\delta_2}\\

\tau_4&=\tfrac{1}{4}{\scriptsize(170-2640 \chi+18872 \chi ^2-80132 \chi ^3+227258 \chi ^4)}\\
&\quad\>\>-\tfrac{1}{4}{\scriptsize(457732\chi^{5}-678443\chi^{6}+754890\chi^{7}-635620\chi^{8}+403736\chi^{9})}\\
&\qquad+\tfrac{1}{4}{\scriptsize(190456\chi^{10}-64576\chi^{11}+14816\chi^{12}-2048\chi^{13}+128\chi^{14})}\\
&\qquad\>\>\>-\tfrac{1}{2}{\scriptsize(85-1150 \chi+7136 \chi ^2-25709 \chi ^3+60351 \chi ^4)\delta_1}\\
&\qquad\quad\>+\tfrac{1}{2}{\scriptsize(97708 \chi ^5-112387 \chi ^6+92774 \chi ^7-54552 \chi ^8)\delta_1}\\
&\qquad\quad\>\>\>+\tfrac{1}{2}{\scriptsize(22232\chi^{9}-5936\chi^{10}+928\chi^{11}-64\chi^{12})\delta_1}\\
&\qquad\qquad\>\,+\tfrac{1}{2}{\scriptsize(40-620 \chi+4080 \chi ^2-15649 \chi ^3+38842 \chi ^4)\delta_2}\\
&\qquad\qquad\quad-\tfrac{1}{2}{\scriptsize(66058\chi^{5}-79507\chi^{6}+68658\chi^{7}-42360\chi^{8})\delta_2}\\
&\qquad\qquad\>\>-\tfrac{1}{2}{\scriptsize(18216 \chi ^9-5168 \chi ^{10}+864 \chi ^{11}-64 \chi ^{12})\delta_2}\\
&\qquad\qquad-\tfrac{1}{2}{\scriptsize(40-540 \chi+3000 \chi ^2-9609 \chi ^3+19604 \chi ^4)\delta_1\delta_2}\\
&\qquad\quad\>\>+\tfrac{1}{2}{\scriptsize(26970 \chi ^5-25770 \chi ^6+17230 \chi ^7-7920 \chi ^8)\delta_1\delta_2}\\
&\qquad\>\>\>\,+\tfrac{1}{2}{\scriptsize(2376 \chi ^9-416 \chi ^{10}+32 \chi ^{11})\delta_1\delta_2}\\

\tau_5&=\tfrac{1}{8}{\scriptsize(20-440 \chi+3984 \chi ^2-21622 \chi ^3+78860 \chi^4)}\\
&\quad\>\>-\tfrac{1}{8}{\scriptsize(205392 \chi ^5-395753 \chi ^6+575934 \chi ^7-639884 \chi ^8)}\\
&\qquad-\tfrac{1}{8}{\scriptsize(543970 \chi ^9-351428 \chi ^{10}+169472 \chi ^{11}-58960 \chi ^{12})}\\
&\qquad\quad-\tfrac{1}{8}{\scriptsize(13920 \chi ^{13}-1984 \chi ^{14}+128 \chi ^{15})}\\
&\qquad\quad\>\>\>-\tfrac{1}{8}{\scriptsize( 20-400 \chi+3184 \chi ^2-15234 \chi ^3+48032 \chi ^4)\delta_1}\\
&\qquad\qquad\>+\tfrac{1}{8}{\scriptsize(105456 \chi ^5-166097 \chi ^6+190412 \chi ^7)\delta_1}\\
&\qquad\qquad\>\>\>\>-\tfrac{1}{8}{\scriptsize(159092 \chi ^8-95656 \chi ^9+40128 \chi ^{10})\delta_1}\\
&\qquad\qquad\,+\tfrac{1}{8}{\scriptsize(11072 \chi ^{11}-1792 \chi ^{12}+128 \chi ^{13})\delta_1}\\
&\qquad\quad\>\>\,+\tfrac{1}{8}{\scriptsize(12-208 \chi +1776 \chi ^2-8938 \chi ^3+29648 \chi ^4)\delta_2}\\
&\qquad\quad-\tfrac{1}{8}{\scriptsize(68304 \chi ^5-112517 \chi ^6+134632 \chi ^7)\delta_2}\\
&\qquad\>\>+\tfrac{1}{8}{\scriptsize(117468 \chi ^8-74008 \chi ^9+32736 \chi ^{10})\delta_2}\\
&\qquad-\tfrac{1}{8}{\scriptsize(9600 \chi ^{11}-1664 \chi ^{12}+128 \chi ^{13})\delta_2}\\
&\qquad\>\>\>-\tfrac{1}{8}{\scriptsize(12-184 \chi+1408 \chi ^2-6110 \chi ^3)\delta_1\delta_2}\\
&\qquad\quad\,-\tfrac{1}{8}{\scriptsize(17268 \chi ^4-33384 \chi ^5+45341 \chi ^6)\delta_1\delta_2}\\
&\qquad\quad\>\>\>\,+\tfrac{1}{8}{\scriptsize(43758 \chi ^7-29920 \chi ^8+14168 \chi ^9)\delta_1\delta_2}\\
&\qquad\qquad\>\>-\tfrac{1}{8}{\scriptsize(4400 \chi ^{10}-800 \chi ^{11}+64 \chi ^{12})\delta_1\delta_2}\\

\end{split}\\
\\

\end{gather*}

青青子衿

青青子衿 发表于 2024-2-10 23:51
\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}
=M\int_{0}^{y_{11}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Vt^{2}\right)}}\\
\\
y_{11}=\frac{x(\frac{1}{M}+\omega_1x^2+\omega_2x^4+\omega_3x^6+\omega_4x^8+\omega_5x^{10})}{1+\tau_1x^2+\tau_2x^4+\tau_3x^6+\tau_4x^8+\tau_5x^{10}}\\
\\
\end{gather*}

\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}
=\frac{1}{\sqrt{11}}\int_{0}^{y_{\scriptsize11}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-V_{11}t^{2})}}\\
\\
y_{\scriptsize11}=\tfrac{\sqrt{\scriptsize11}\,x(1+\tau_{\scriptsize11,1}x^2+\tau_{\scriptsize11,2}x^4+\tau_{\scriptsize11,3}x^6+\tau_{\scriptsize11,4}x^8+\tau_{\scriptsize11,5}x^{10})}{1+\omega_{\scriptsize11,1}x^2+\omega_{\scriptsize11,2}x^4+\omega_{\scriptsize11,3}x^6+\omega_{\scriptsize11,4}x^8+\omega_{\scriptsize11,5}x^{10}}\\
\\
\begin{split}
U_{\scriptsize11}&={\scriptsize\left(\tfrac{3\sqrt{2}+\sqrt{22}}{12}
-\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}-\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}\\
V_{\scriptsize11}&={\scriptsize\left(\tfrac{\sqrt{22}-3 \sqrt{2}}{12}+\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}\\
\end{split}\\
\\
\qquad\begin{split}
\tau_{\scriptsize11,1}&={\scriptsize-\tfrac{5}{2}+\tfrac{(7\sqrt{3}+3\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}}\\

\tau_{\scriptsize11,2}&={\scriptsize4-\tfrac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}}\\

\tau_{\scriptsize11,3}&={\scriptsize-\tfrac{14-\sqrt{11}}{4}+\tfrac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}+\tfrac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33})(17+3\sqrt{33})^{1/3}}{48}}\\

\tau_{\scriptsize11,4}&={\scriptsize\tfrac{101-12\sqrt{11}}{48}-\tfrac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}-\tfrac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33})^{1/3}}{96}}\\

\tau_{\scriptsize11,5}&={\scriptsize-\tfrac{583-30\sqrt{11}}{1056}+\tfrac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}+\tfrac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33})(17+3\sqrt{33})^{1/3}}{192}}\\

\omega_{\scriptsize11,1}&={\scriptsize-\tfrac{\sqrt{11}}{6}+\tfrac{(11\sqrt{3}+5\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(11\sqrt{3}-5\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\omega_{\scriptsize11,2}&={\scriptsize\tfrac{5\sqrt{11}}{6}-\tfrac{(33+66\sqrt{3}+34\sqrt{11}-\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(33-66\sqrt{3}+34\sqrt{11}+\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\omega_{\scriptsize11,3}&={\scriptsize\tfrac{\sqrt{11}}{6}+\tfrac{(264+121\sqrt{3}+77\sqrt{11}+24\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(264-121\sqrt{3}+77\sqrt{11}-24\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\omega_{\scriptsize11,4}&={\scriptsize-\tfrac{55+30\sqrt{11}\>\>\!\!}{48}-\tfrac{(407+132\sqrt{3}+108\sqrt{11}+37\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(407-132\sqrt{3}+108\sqrt{11}-37\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\omega_{\scriptsize11,5}&={\scriptsize\tfrac{44+33\sqrt{11}\>\>\!\!}{96}+\tfrac{(220+33\sqrt{3}+45\sqrt{11}+20\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(220-33\sqrt{3}+45\sqrt{11}-20\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\
\end{split}

\end{gather*}
【13阶奇异模】
\begin{gather*}
I_{13}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{13}t^{2})}}
=\frac{1}{\sqrt{13}}\int_{0}^{y_{\scriptsize13}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-V_{13}t^{2})}}\\
\\
y_{\scriptsize13}=\tfrac{\sqrt{\scriptsize13}\,x(1+\tau_{\scriptsize13,1}x^2+\tau_{\scriptsize13,2}x^4+\tau_{\scriptsize13,3}x^6+\tau_{\scriptsize13,4}x^8+\tau_{\scriptsize13,5}x^{10}+\tau_{\scriptsize13,6}x^{12})}{1+\omega_{\scriptsize13,1}x^2+\omega_{\scriptsize13,2}x^4+\omega_{\scriptsize13,3}x^6+\omega_{\scriptsize13,4}x^8+\omega_{\scriptsize13,5}x^{10}+\omega_{\scriptsize13,6}x^{12}}\\
\\
\begin{split}
U_{\scriptsize13}&={\scriptsize\left(\tfrac{\sqrt{26}-5\sqrt{2}+2\sqrt{5\sqrt{13}-17}}{4}\right)^2}\\
V_{\scriptsize13}&={\scriptsize\left(\tfrac{5\sqrt{2}-\sqrt{26}+2\sqrt{5\sqrt{13}-17}}{4}\right)^2}\\
\end{split}\quad\\
\\
\qquad\>\>\>\begin{split}
\tau_{\scriptsize13,1}&={\scriptsize\tfrac{\sqrt{10\sqrt{13}+34}-6}{2}}\\
\tau_{\scriptsize13,2}&={\scriptsize\tfrac{623-161\sqrt{13}-5\sqrt{10\sqrt{13}+34}}{4}}\\
\tau_{\scriptsize13,3}&={\scriptsize\tfrac{161\sqrt{13}-613+8\sqrt{4993\sqrt{13}-17986}}{2}}\\
\tau_{\scriptsize13,4}&={\scriptsize\tfrac{6232-1713\sqrt{13}-\sqrt{2884138\sqrt{13}-10395806}}{4}}\\
\tau_{\scriptsize13,5}&={\scriptsize\tfrac{3104\sqrt{13}-11242+\sqrt{52826458\sqrt{13}-190465966}}{8}}\\
\tau_{\scriptsize13,6}&={\scriptsize\tfrac{130351-36085\sqrt{13}-13\sqrt{37646890\sqrt{13}-135737438}}{208}}\\
\end{split}
\qquad\begin{split}
\omega_{\scriptsize13,1}&={\scriptsize\tfrac{\sqrt{962\sqrt{13}-3354}}{2}}\\
\omega_{\scriptsize13,2}&={\scriptsize\tfrac{343\sqrt{13}-1209-\sqrt{141466\sqrt{13}-509678}}{4}}\\
\omega_{\scriptsize13,3}&={\scriptsize\tfrac{126\sqrt{13}-468+\sqrt{6602258\sqrt{13}-23804586}}{2}}\\
\omega_{\scriptsize13,4}&={\scriptsize\tfrac{27092-7509\sqrt{13}-3\sqrt{48941698\sqrt{13}-176461766}}{4}}\\
\omega_{\scriptsize13,5}&={\scriptsize\tfrac{3494\sqrt{13}-12610+3\sqrt{3052010\sqrt{13}-11004162}}{8}}\\
\omega_{\scriptsize13,6}&={\scriptsize\tfrac{30695\sqrt{13}-110669-\sqrt{8975786170\sqrt{13}-32362657262}}{16}}\\
\end{split}
\end{gather*}

1. \tau_{1}=-\frac{5}{2}+\frac{(7\sqrt{3}+3\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\frac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}
2. \tau_{2}=4-\frac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
3. \tau_{3}=-\frac{14-\sqrt{11}}{4}+\frac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}+\frac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33})(17+3\sqrt{33})^{1/3}}{48}
4. \tau_{4}=\frac{101-12\sqrt{11}}{48}-\frac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}-\frac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33})^{1/3}}{96}
5. \tau_{5}=-\frac{583-30\sqrt{11}}{1056}+\frac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}+\frac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33})(17+3\sqrt{33})^{1/3}}{192}
6. \omega_{1}=-\frac{\sqrt{11}}{6}+\frac{(11\sqrt{3}+5\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\frac{(11\sqrt{3}-5\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}
7. \omega_{2}=\frac{5\sqrt{11}}{6}-\frac{(33+66\sqrt{3}+34\sqrt{11}-\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(33-66\sqrt{3}+34\sqrt{11}+\sqrt{33})(17+3\sqrt{33})^{1/3}}{24}
8. \omega_{3}=\frac{\sqrt{11}}{6}+\frac{(264+121\sqrt{3}+77\sqrt{11}+24\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}-\frac{(264-121\sqrt{3}+77\sqrt{11}-24\sqrt{33})(17+3\sqrt{33})^{1/3}}{48}
9. \omega_{4}=-\frac{55+30\sqrt{11}}{48}-\frac{(407+132\sqrt{3}+108\sqrt{11}+37\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}+\frac{(407-132\sqrt{3}+108\sqrt{11}-37\sqrt{33})(17+3\sqrt{33})^{1/3}}{96}
10. \omega_{5}=\frac{44+33\sqrt{11}}{96}+\frac{(220+33\sqrt{3}+45\sqrt{11}+20\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}-\frac{(220-33\sqrt{3}+45\sqrt{11}-20\sqrt{33})(17+3\sqrt{33})^{1/3}}{192}
11. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt
12. \frac{1}{\sqrt{11}}\int_{0}^{f\left(x\right)}\frac{1}{\sqrt{(1-t^{2})(1-l^{2}t^{2})}}dt
13. k=\frac{3\sqrt{2}+\sqrt{22}}{12}-\frac{(\sqrt{2}+\sqrt{6})(\sqrt{3}+\sqrt{11})(3\sqrt{33}-17)^{1/3}}{24}-\frac{(\sqrt{6}-\sqrt{2})(\sqrt{11}-\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
14. l=\frac{\sqrt{22}-3\sqrt{2}}{12}+\frac{(\sqrt{6}-\sqrt{2})(\sqrt{3}+\sqrt{11})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(\sqrt{2}+\sqrt{6})(\sqrt{11}-\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
15. f\left(x\right)=\frac{\sqrt{11}x(1+\tau_{1}x^{2}+\tau_{2}x^{4}+\tau_{3}x^{6}+\tau_{4}x^{8}+\tau_{5}x^{10})}{1+\omega_{1}x^{2}+\omega_{2}x^{4}+\omega_{3}x^{6}+\omega_{4}x^{8}+\omega_{5}x^{10}}
16. \int_{0}^{x}\frac{g_{11}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt-\sqrt{11}\int_{0}^{x}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt+\int_{0}^{f\left(x\right)}\sqrt{\frac{1-l^{2}t^{2}}{1-t^{2}}}dt
17. g_{11}=\frac{(5+\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{3}+\frac{(\sqrt{33}-5)\sqrt[3]{17+3\sqrt{33}}}{3}-\frac{1}{3}
18. \frac{h_{11}x(1+\sigma_{1}x^{2}+\sigma_{2}x^{4}+\sigma_{3}x^{6}+\sigma_{4}x^{8})\sqrt{(1-x^{2})(1-k^{2}x^{2})}}{1+\omega_{1}x^{2}+\omega_{2}x^{4}+\omega_{3}x^{6}+\omega_{4}x^{8}+\omega_{5}x^{10}}
19. h_{11}=\frac{(5+\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{3}+\frac{(\sqrt{33}-5)\sqrt[3]{17+3\sqrt{33}}}{3}-\frac{1}{3}
20. \int_{0}^{x}\frac{g_{11}+\sqrt{11}(k^{2}t^{2}-l^{2}f\left(t\right)^{2})}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt
21. \frac{h_{11}x(1+\sigma_{1}x^{2}+\sigma_{2}x^{4}+\sigma_{3}x^{6}+\sigma_{4}x^{8})\sqrt{(1-x^{2})(1-k^{2}x^{2})}}{1+\omega_{1}x^{2}+\omega_{2}x^{4}+\omega_{3}x^{6}+\omega_{4}x^{8}+\omega_{5}x^{10}}
22. \sigma_{1}=-\frac{126-4\sqrt{11}}{39}+\frac{(66+89\sqrt{3}+53\sqrt{11}+14\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{78}-\frac{(66-89\sqrt{3}+53\sqrt{11}-14\sqrt{33})\sqrt[3]{17+3\sqrt{33}}}{78}
23. \sigma_{2}=\frac{46+8\sqrt{11}}{13}-\frac{(331+328\sqrt{3}+200\sqrt{11}+45\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{104}+\frac{(331-328\sqrt{3}+200\sqrt{11}-45\sqrt{33})\sqrt[3]{17+3\sqrt{33}}}{104}
24. \sigma_{3}=-\frac{142+41\sqrt{11}}{78}+\frac{(712+469\sqrt{3}+305\sqrt{11}+88\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{156}-\frac{(712-469\sqrt{3}+305\sqrt{11}-88\sqrt{33})\sqrt[3]{17+3\sqrt{33}}}{156}
25. \sigma_{4}=\frac{100\sqrt{11}-95}{624}-\frac{5(437+244\sqrt{3}+164\sqrt{11}+47\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{1248}+\frac{5(437-244\sqrt{3}+164\sqrt{11}-47\sqrt{33})\sqrt[3]{17+3\sqrt{33}}}{1248}

复制代码

\begin{align*}
\left\{
\begin{split}
\mathbf{v}_0&\in W\mathrm{~chosen},\\
\mathbf{w}_k&=\mathrm{Pr}_W(\mathbf{v}_k-\alpha_k\nabla_w\Phi(\mathbf{v}_k,\mathbf{v}_k)),\\
C_k&=\{z\in \boldsymbol{W}:\lVert\boldsymbol{z}-\mathbf{w}_k\rVert\cdot\lVert\mathbf{w}_k-\mathbf{v}_k\rVert\leqslant
\left| \langle\boldsymbol{z}-\mathbf{v}_k,\mathbf{w}_k-\mathbf{v}_k\rangle\right|\},\\
Q_k&=\{\boldsymbol{z}\in\boldsymbol{W}:\langle\boldsymbol{z}-\mathbf{v}_k,\mathbf{v}_k-\mathbf{v}_0\rangle\geqslant0\},\\
\mathbf{v}_{k+1}&=\mathrm{Pr}_{C_k\cap Q_k}\mathbf{v}_0,\mathrm{~}k=0,1,2...
\end{split}\right.
\end{align*}



\begin{gather*}
\int_{0}^{1}\frac{g_{11}}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}{\mathrm{d}}t=\sqrt{11}\int_{0}^{1}\sqrt{\frac{1-U_{11}t^{2}}{1-t^{2}}}{\mathrm{d}}t-\int_{0}^{1}\sqrt{\frac{1-V_{11}t^{2}}{1-t^{2}}}{\mathrm{d}}t\\
\\
\begin{split}
g_{\scriptsize11}&=\tfrac{(5+\sqrt{33})\sqrt[3]{3\sqrt{33}-17}}{3}+\tfrac{(\sqrt{33}-5)\sqrt[3]{17+3\sqrt{33}}}{3}-\tfrac{1}{3}\\
U_{\scriptsize11}&=\tfrac{6+\sqrt{11}}{12}-\tfrac{(4\sqrt{3}+2\sqrt{11})\sqrt[3]{3\sqrt{33}-17}}{12}-\tfrac{(4\sqrt{3}-2\sqrt{11})\sqrt[3]{17+3\sqrt{33}}}{12}\\
V_{\scriptsize11}&=\tfrac{6-\sqrt{11}}{12}+\tfrac{(4\sqrt{3}+2\sqrt{11})\sqrt[3]{3\sqrt{33}-17}}{12}+\tfrac{(4\sqrt{3}-2\sqrt{11})\sqrt[3]{17+3\sqrt{33}}}{12}\\

\end{split}
\end{gather*}

\begin{gather*}
\begin{split}
\Xi_{11}(x)&=\int_{0}^{x}{\small\frac{g_{11}}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}}{\mathrm{d}}t\\
&\quad\>\>\>\>{\small-\,\sqrt{11}}\int_{0}^{x}\sqrt{\small\frac{1-U_{11}t^{2}}{1-t^{2}}}{\mathrm{d}}t\\
&\qquad\>\>\>\>+\int_{0}^{y_{\scriptsize11}(x)}\sqrt{\small\frac{1-V_{11}t^{2}}{1-t^{2}}}{\mathrm{d}}t\\
&=\rho_{\scriptsize11}(x)\sqrt{\small(1-x^{2})(1-U_{11}x^{2})}
\end{split}\qquad\\
\\
\begin{split}
y_{\scriptsize11}(x)
&=\tfrac{\sqrt{\scriptsize11}\,x(1+\tau_{\scriptsize11,1}x^2+\tau_{\scriptsize11,2}x^4+\tau_{\scriptsize11,3}x^6+\tau_{\scriptsize11,4}x^8+\tau_{\scriptsize11,5}x^{10})}

{1+\omega_{\scriptsize11,1}x^2+\omega_{\scriptsize11,2}x^4+\omega_{\scriptsize11,3}x^6+\omega_{\scriptsize11,4}x^8+\omega_{\scriptsize11,5}x^{10}}\\
\rho_{\scriptsize11}(x)
&=\tfrac{h_{11}x(1+\sigma_{\scriptsize11,1}x^{2}+\sigma_{\scriptsize11,2}x^{4}+\sigma_{\scriptsize11,3}x^{6}+\sigma_{\scriptsize11,4}x^{8})}

{1+\omega_{\scriptsize11,1}x^2+\omega_{\scriptsize11,2}x^4+\omega_{\scriptsize11,3}x^6+\omega_{\scriptsize11,4}x^8+\omega_{\scriptsize11,5}x^{10}}
\end{split}\\
\\
\\
\begin{split}
g_{\scriptsize11}&=h_{11}={\scriptsize\tfrac{(5+\sqrt{33})(3\sqrt{33}-17)^{1/3}}{3}+\tfrac{(\sqrt{33}-5)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{3}-\tfrac{1}{3}}
\\
U_{\scriptsize11}&={\scriptsize\tfrac{6+\sqrt{11}}{12}-\tfrac{(4\sqrt{3}+2\sqrt{11})(3\sqrt{33}-17)^{1/3}}{12}-\tfrac{(4\sqrt{3}-2\sqrt{11})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{12}}
\\
V_{\scriptsize11}&={\scriptsize\tfrac{6-\sqrt{11}}{12}+\tfrac{(4\sqrt{3}+2\sqrt{11})(3\sqrt{33}-17)^{1/3}}{12}+\tfrac{(4\sqrt{3}-2\sqrt{11})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{12}}
\\
\end{split}\\
\\
\qquad\quad\begin{split}
\tau_{\scriptsize11,1}&={\scriptsize-\tfrac{5}{2}+\tfrac{(7\sqrt{3}+3\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}}\\

\tau_{\scriptsize11,2}&={\scriptsize4-\tfrac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}}\\

\tau_{\scriptsize11,3}&={\scriptsize-\tfrac{14-\sqrt{11}}{4}+\tfrac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}+\tfrac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33})(17+3\sqrt{33})^{1/3}}{48}}\\

\tau_{\scriptsize11,4}&={\scriptsize\tfrac{101-12\sqrt{11}}{48}-\tfrac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}-\tfrac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33})^{1/3}}{96}}\\

\tau_{\scriptsize11,5}&={\scriptsize-\tfrac{583-30\sqrt{11}}{1056}+\tfrac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}+\tfrac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33})(17+3\sqrt{33})^{1/3}}{192}}\\

\omega_{\scriptsize11,1}&={\scriptsize-\tfrac{\sqrt{11}}{6}+\tfrac{(11\sqrt{3}+5\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(11\sqrt{3}-5\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\omega_{\scriptsize11,2}&={\scriptsize\tfrac{5\sqrt{11}}{6}-\tfrac{(33+66\sqrt{3}+34\sqrt{11}-\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(33-66\sqrt{3}+34\sqrt{11}+\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\omega_{\scriptsize11,3}&={\scriptsize\tfrac{\sqrt{11}}{6}+\tfrac{(264+121\sqrt{3}+77\sqrt{11}+24\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(264-121\sqrt{3}+77\sqrt{11}-24\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\omega_{\scriptsize11,4}&={\scriptsize-\tfrac{55+30\sqrt{11}\>\>\!\!}{48}-\tfrac{(407+132\sqrt{3}+108\sqrt{11}+37\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(407-132\sqrt{3}+108\sqrt{11}-37\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\omega_{\scriptsize11,5}&={\scriptsize\tfrac{44+33\sqrt{11}\>\>\!\!}{96}+\tfrac{(220+33\sqrt{3}+45\sqrt{11}+20\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(220-33\sqrt{3}+45\sqrt{11}-20\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\

\sigma_{\scriptsize11,1}&={\scriptsize-\tfrac{126-4\sqrt{11}}{39}+\tfrac{(66+89\sqrt{3}+53\sqrt{11}+14\sqrt{33})(3\sqrt{33}-17)^{1/3}}{78}-\tfrac{(66-89\sqrt{3}+53\sqrt{11}-14\sqrt{33})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{78}}\\

\sigma_{\scriptsize11,2}&={\scriptsize\tfrac{46+8\sqrt{11}}{13}-\tfrac{(331+328\sqrt{3}+200\sqrt{11}+45\sqrt{33})(3\sqrt{33}-17)^{1/3}}{104}+\tfrac{(331-328\sqrt{3}+200\sqrt{11}-45\sqrt{33})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{104}}\\

\sigma_{\scriptsize11,3}&={\scriptsize-\tfrac{142+41\sqrt{11}}{78}+\tfrac{(712+469\sqrt{3}+305\sqrt{11}+88\sqrt{33})(3\sqrt{33}-17)^{1/3}}{156}-\tfrac{(712-469\sqrt{3}+305\sqrt{11}-88\sqrt{33})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{156}}\\

\sigma_{\scriptsize11,4}&={\scriptsize\tfrac{100\sqrt{11}-95}{624}-\tfrac{5(437+244\sqrt{3}+164\sqrt{11}+47\sqrt{33})(3\sqrt{33}-17)^{1/3}}{1248}+\tfrac{5(437-244\sqrt{3}+164\sqrt{11}-47\sqrt{33})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{1248}}\\

\end{split}
\end{gather*}

青青子衿

青青子衿 发表于 2024-2-28 06:03
\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}
=\frac{1}{\sqrt{11}}\int_{0}^{y_{\scriptsize11}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-V_{11}t^{2})}}\\
\\
y_{\scriptsize11}=\tfrac{\sqrt{\scriptsize11}\,x(1+\tau_{\scriptsize11,1}x^2+\tau_{\scriptsize11,2}x^4+\tau_{\scriptsize11,3}x^6+\tau_{\scriptsize11,4}x^8+\tau_{\scriptsize11,5}x^{10})}{1+\omega_{\scriptsize11,1}x^2+\omega_{\scriptsize11,2}x^4+\omega_{\scriptsize11,3}x^6+\omega_{\scriptsize11,4}x^8+\omega_{\scriptsize11,5}x^{10}}\\
\\
\end{gather*}

[Submitted on 22 Jun 2020 (v1), last revised 2 Jul 2020 (this version, v2)]
Some singular values of the elliptic lambda function and incredible cubic identities
Genki Shibukawa
arxiv.org/abs/2006.12034


\begin{align*}
\frac{\scriptsize12\Big(1+240\sum\limits_{\scriptsize{k=1}}^{\scriptsize+\infty}\frac{k^{3}e^{-k\pi}}{1-e^{-k\pi}}\Big)}{\scriptsize\Big[\Big(1+240\sum\limits_{\scriptsize{k=1}}^{\scriptsize+\infty}\frac{k^{3}e^{-k\pi}}{1-e^{-k\pi}}\Big)\!\raise{4pt}^{3}-\Big(1-504\sum\limits_{\scriptsize{k=1}}^{\scriptsize+\infty}\frac{k^{5}e^{-k\pi}}{1-e^{-k\pi}}\Big)\!\raise{4pt}^{2}\>\Big]^{1/3}}=66
\end{align*}

青青子衿

青青子衿 发表于 2024-8-7 09:41
[Submitted on 22 Jun 2020 (v1), last revised 2 Jul 2020 (this version, v2)]

\begin{gather*}
\eta ^2+\eta =\xi ^3-\xi ^2-7820 \xi -263580\\
\\
\begin{split}
&\left\{\tiny
\begin{split}
X&=\frac{\xi ^5+210 \xi ^4+16645 \xi ^3+596100 \xi ^2+8522055 \xi +16477645}{\left(5 \xi ^2+505 \xi +12751\right)^2}\\
Y&=\frac{\eta -62}{125}+\frac{(2 \eta +1) \left(17025 \xi ^4+3461925 \xi ^3+263983830 \xi ^2+8946483305 \xi +113698972812\right)}{125 \left(5 \xi ^2+505 \xi +12751\right)^3}
\end{split}
\right.\\
&\left\{\tiny
\begin{split}
\xi&=\frac{X^5-42 X^4+2163 X^3-30420 X^2+170498 X-324599}{(16-X)^2 (5-X)^2}\\
\eta&=Y-\frac{11(2 Y+1)(71 X^4-969 X^3+5337 X^2-18857 X+22969)}{(16-X)^3 (5-X)^3}
\end{split}
\right.\\
\\
\end{split}
\\
Y^2 + Y = X^3 - X^2 - 10 X - 20\\
\\
\begin{split}
&\left\{\tiny
\begin{split}
X&=\frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}\\
Y&=y-\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}
\end{split}
\right.\\
&\left\{\tiny
\begin{split}
x&=\frac{(X-5) \left(X^4+15 X^3+120 X^2+200 X+155\right)}{\left(5 X^2+5 X-29\right)^2}\\
y&=\frac{Y-62}{125}-\frac{11(2 Y+1)\left(225 X^4-3675 X^3-9480 X^2-48205 X-37762\right)}{125 \left(5 X^2+5 X-29\right)^3}
\end{split}
\right.\\
\end{split}\\
\\
y^2+y=x^3-x^2\\
\end{gather*}

1. Y^2 + Y - (X^3 - X^2 - 10 X - 20) /. {X -> (
2.     1 - 2 x + 3 x^3 - 2 x^4 + x^5)/((1 - x)^2 x^2),
3.    Y -> y - ( (1 + 2 y) (1 - 3 x + 3 x^2 - x^3 - x^4))/((1 -
4.         x)^3 x^3)} // Factor
5. y^2 + y - (x^3 -
6.      x^2) /. {x -> ((X - 5) (155 + 200 X + 120 X^2 + 15 X^3 +
7.        X^4))/(5 X^2 + 5 X - 29)^2,
8.    y -> (Y - 62)/125 - (
9.      11 (2 Y + 1) (225 X^4 - 3675 X^3 - 9480 X^2 - 48205 X -
10.         37762) )/(125 (5 X^2 + 5 X - 29)^3)} // Factor
11. Y^2 + Y - (X^3 - X^2 - 10 X - 20) /. {X -> (
12.     16477645 + 8522055 \[Xi] + 596100 \[Xi]^2 + 16645 \[Xi]^3 +
13.      210 \[Xi]^4 + \[Xi]^5)/(12751 + 505 \[Xi] + 5 \[Xi]^2)^2,
14.    Y -> (\[Eta] - 62)/
15.      125 + ((1 + 2 \[Eta]) (113698972812 + 8946483305 \[Xi] +
16.         263983830 \[Xi]^2 + 3461925 \[Xi]^3 + 17025 \[Xi]^4) )/(
17.      125 (12751 + 505 \[Xi] + 5 \[Xi]^2)^3)} // Factor
18. \[Eta]^2 + \[Eta] - (\[Xi]^3 - \[Xi]^2 - 7820 \[Xi] -
19.      263580) /. {\[Xi] -> (
20.     X^5 - 42 X^4 + 2163 X^3 - 30420 X^2 + 170498 X -
21.      324599)/((16 - X)^2 (5 - X)^2),
22.    \[Eta] ->
23.     Y - (11  (1 + 2 Y) (71 X^4 - 969 X^3 + 5337 X^2 - 18857 X +
24.         22969))/((16 - X)^3 (5 - X)^3)} // Factor

复制代码

\begin{gather*}
y^2=x^3-336 x+2576\\
\\
\begin{split}
&\left\{\tiny
\begin{split}
X&=\frac{
\begin{split}
&\left(x^{13}-168 x^{12}+35280 x^{11}-1907584 x^{10}+25853184 x^9+768144384 x^8-26402009088 x^7\right.\\
&\qquad+200488845312 x^6+1144060968960 x^5-8715049369600 x^4-303633848598528 x^3\\
&\qquad\qquad\left.+4285571870490624 x^2-21785403233140736 x+42401094973784064\right)
\end{split}
}{\left(x^3-336 x-448\right)^2 \left(x^3-84 x^2+1344 x-5824\right)^2}\\
\\
Y&=\frac{
\begin{split}
&\left(x^9-336 x^8+23856 x^7-885696 x^6+22541568 x^5-402511872 x^4+4963811328 x^3\right.\\
&\qquad\left.-39770701824 x^2+180392755200 x-324864311296\right)\left(x^9+84 x^8+2352 x^7-978432 x^5\right.\\
&\qquad\qquad\left.-39588864 x^4+1014558720 x^3-2613166080 x^2-82272583680 x+528612589568\right)y
\end{split}
}{\left(x^3-336 x-448\right)^3 \left(x^3-84 x^2+1344 x-5824\right)^3}\\
\end{split}
\right.\\
\\
&\left\{\tiny
\begin{split}
x&=\frac{
\begin{split}
&\left(X^{13}+2184 X^{12}+3699696 X^{11}+2975423360 X^{10}+801676302336 X^9-346198983975936 X^8-366859624464592896 X^7\right.\\
&\qquad-148256947508460650496 X^6-37090882017886975229952 X^5-6523518309214134130180096 X^4\\
&\qquad\qquad-848935683123809074972459008 X^3-80065347092339317917679091712 X^2\\
&\qquad\qquad\qquad-4850696291121055411852135104512 X-138134742021887241348250117079040)
\end{split}
}{\left(13 X^6+14196 X^5+3992352 X^4-330005312 X^3-322363501824 X^2-50110062213120 X-2444790664859648\right)^2}\\
\\
y&=\frac{
\begin{split}
&\left(X^{18}+3276 X^{17}+2605680 X^{16}+594386688 X^{15}+447725327616 X^{14}+1391652517764096 X^{13}+1805645717703389184 X^{12}\right.\\
&\qquad+1454352220829946396672 X^{11}+838660048536109377650688 X^{10}+362190048106806033257070592 X^9\\
&\qquad\qquad+5413756987876403708663354395459584 X^6+728411557024185952172036991498584064 X^5\\
&\qquad\qquad\qquad+67923123493588476094342985916497264640 X^4+3932504432370976278995731975111150927872 X^3\\
&\qquad\qquad+97064352276673717766275863807684683759616 X^2-2284697174750057440708540443744645106630656 X\\
&\qquad-152688001724463750104335674945274645418344448)Y
\end{split}
}{\left(13 X^6+14196 X^5+3992352 X^4-330005312 X^3-322363501824 X^2-50110062213120 X-2444790664859648\right)^3}\\
\end{split}
\right.\\
\\
\end{split}
\\
Y^2=X^3-131376 X-18343024\\
\end{gather*}

青青子衿

青青子衿 发表于 2024-2-28 06:03
\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}
=\frac{1}{\sqrt{11}}\int_{0}^{y_{\scriptsize11}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-V_{11}t^{2})}}\\
\\
y_{\scriptsize11}=\tfrac{\sqrt{\scriptsize11}\,x(1+\tau_{\scriptsize11,1}x^2+\tau_{\scriptsize11,2}x^4+\tau_{\scriptsize11,3}x^6+\tau_{\scriptsize11,4}x^8+\tau_{\scriptsize11,5}x^{10})}{1+\omega_{\scriptsize11,1}x^2+\omega_{\scriptsize11,2}x^4+\omega_{\scriptsize11,3}x^6+\omega_{\scriptsize11,4}x^8+\omega_{\scriptsize11,5}x^{10}}\\
\\
\begin{split}
U_{\scriptsize11}&={\scriptsize\left(\tfrac{3\sqrt{2}+\sqrt{22}}{12}
-\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}-\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}\\
V_{\scriptsize11}&={\scriptsize\left(\tfrac{\sqrt{22}-3 \sqrt{2}}{12}+\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}\\
\end{split}\\
\\
\qquad\begin{split}
\tau_{\scriptsize11,1}&={\scriptsize-\tfrac{5}{2}+\tfrac{(7\sqrt{3}+3\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}}\\

\tau_{\scriptsize11,2}&={\scriptsize4-\tfrac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}}\\

\tau_{\scriptsize11,3}&={\scriptsize-\tfrac{14-\sqrt{11}}{4}+\tfrac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}+\tfrac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33})(17+3\sqrt{33})^{1/3}}{48}}\\

\tau_{\scriptsize11,4}&={\scriptsize\tfrac{101-12\sqrt{11}}{48}-\tfrac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}-\tfrac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33})^{1/3}}{96}}\\

\tau_{\scriptsize11,5}&={\scriptsize-\tfrac{583-30\sqrt{11}}{1056}+\tfrac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}+\tfrac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33})(17+3\sqrt{33})^{1/3}}{192}}\\

\omega_{\scriptsize11,1}&={\scriptsize-\tfrac{\sqrt{11}}{6}+\tfrac{(11\sqrt{3}+5\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(11\sqrt{3}-5\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\omega_{\scriptsize11,2}&={\scriptsize\tfrac{5\sqrt{11}}{6}-\tfrac{(33+66\sqrt{3}+34\sqrt{11}-\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(33-66\sqrt{3}+34\sqrt{11}+\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\omega_{\scriptsize11,3}&={\scriptsize\tfrac{\sqrt{11}}{6}+\tfrac{(264+121\sqrt{3}+77\sqrt{11}+24\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(264-121\sqrt{3}+77\sqrt{11}-24\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\omega_{\scriptsize11,4}&={\scriptsize-\tfrac{55+30\sqrt{11}\>\>\!\!}{48}-\tfrac{(407+132\sqrt{3}+108\sqrt{11}+37\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(407-132\sqrt{3}+108\sqrt{11}-37\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\omega_{\scriptsize11,5}&={\scriptsize\tfrac{44+33\sqrt{11}\>\>\!\!}{96}+\tfrac{(220+33\sqrt{3}+45\sqrt{11}+20\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(220-33\sqrt{3}+45\sqrt{11}-20\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\
\end{split}

\end{gather*}

\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}
=\frac{1}{i\sqrt{11}}\int_{0}^{\hat{y}_{\scriptsize11}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}\\
\\
\hat{y}_{\scriptsize11}=\tfrac{i\sqrt{\scriptsize11}\,x(1+\tau_{\scriptsize11,1}x^2+\tau_{\scriptsize11,2}x^4+\tau_{\scriptsize11,3}x^6+\tau_{\scriptsize11,4}x^8+\tau_{\scriptsize11,5}x^{10})}{\sqrt{1-x^2}(1+\hat\omega_{\scriptsize11,1}x^2+\hat\omega_{\scriptsize11,2}x^4+\hat\omega_{\scriptsize11,3}x^6+\hat\omega_{\scriptsize11,4}x^8+\hat\omega_{\scriptsize11,5}x^{10})}\\
\\
\begin{split}
U_{\scriptsize11}&={\scriptsize\left(\tfrac{3\sqrt{2}+\sqrt{22}}{12}
-\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}-\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}

\end{split}\\
\\
\qquad\begin{split}
\tau_{\scriptsize11,1}&={\scriptsize-\tfrac{5}{2}+\tfrac{(7\sqrt{3}+3\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\tau_{\scriptsize11,2}&={\scriptsize4-\tfrac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\tau_{\scriptsize11,3}&={\scriptsize-\tfrac{14-\sqrt{11}}{4}+\tfrac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}+\tfrac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\tau_{\scriptsize11,4}&={\scriptsize\tfrac{101-12\sqrt{11}}{48}-\tfrac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}-\tfrac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\tau_{\scriptsize11,5}&={\scriptsize-\tfrac{583-30\sqrt{11}}{1056}+\tfrac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}+\tfrac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\

\hat\omega_{\scriptsize11,1}&={\scriptsize-\tfrac{30+\sqrt{11}}{6}+\tfrac{(11\sqrt{3}+5\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(11\sqrt{3}-5\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\hat\omega_{\scriptsize11,2}&={\scriptsize\tfrac{60-\sqrt{11}}{6}-\tfrac{(33+110\sqrt{3}+46\sqrt{11}-\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(33-110\sqrt{3}+46\sqrt{11}+\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\hat\omega_{\scriptsize11,3}&={\scriptsize-\tfrac{5(6-\sqrt{11}\>\>\!\!)}{3}+\tfrac{(253\sqrt{3}+113\sqrt{11}-66-30\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(113\sqrt{11}+30\sqrt{33}-66-253\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\hat\omega_{\scriptsize11,4}&={\scriptsize-\tfrac{37(2\sqrt{11}-5)}{48}-\tfrac{(264\sqrt{3}+112\sqrt{11}-253-71\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(112\sqrt{11}+71\sqrt{33}-253-264\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\hat\omega_{\scriptsize11,5}&={\scriptsize\tfrac{53\sqrt{11}-30}{96}-\tfrac{(198+50\sqrt{33}-77\sqrt{3}-25\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(25\sqrt{11}+50\sqrt{33}-198-77\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\
\end{split}

\end{gather*}




\begin{gather*}
I_{1/11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-V_{11}t^{2})}}
=\frac{1}{i\sqrt{11}}\int_{0}^{\hat{\hat{y}}_{\scriptsize11}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-V_{11}t^{2})}}\\
\\
\hat{\hat{y}}_{\scriptsize11}=\tfrac{i\sqrt{\scriptsize11}\,x(1+\hat{\hat{\tau}}_{\scriptsize11,1}x^2+\hat{\hat{\tau}}_{\scriptsize11,2}x^4+\hat{\hat{\tau}}_{\scriptsize11,3}x^6+\hat{\hat{\tau}}_{\scriptsize11,4}x^8+\hat{\hat{\tau}}_{\scriptsize11,5}x^{10})}{\sqrt{1-x^2}(1+\hat{\hat{\omega}}_{\scriptsize11,1}x^2+\hat{\hat{\omega}}_{\scriptsize11,2}x^4+\hat{\hat{\omega}}_{\scriptsize11,3}x^6+\hat{\hat{\omega}}_{\scriptsize11,4}x^8+\hat{\hat{\omega}}_{\scriptsize11,5}x^{10})}\\
\\
\begin{split}
V_{\scriptsize11}&={\scriptsize\left(\tfrac{\sqrt{22}-3 \sqrt{2}}{12}+\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}\\

\end{split}\\
\\
\qquad\begin{split}
\hat{\hat{\tau}}_{\scriptsize11,1}&={\scriptsize-\tfrac{5}{2}-\tfrac{(7\sqrt{3}+3\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}-\tfrac{(7\sqrt{3}-3\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\hat{\hat{\tau}}_{\scriptsize11,2}&={\scriptsize4+\tfrac{(15+56\sqrt{3}+24\sqrt{11}+5\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(56\sqrt{3}+5\sqrt{33}-15-24\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\hat{\hat{\tau}}_{\scriptsize11,3}&={\scriptsize-\tfrac{14+\sqrt{11}}{4}-\tfrac{(45+106\sqrt{3}+42\sqrt{11}+15\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(106\sqrt{3}+15\sqrt{33}-45-42\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\hat{\hat{\tau}}_{\scriptsize11,4}&={\scriptsize\tfrac{101+12\sqrt{11}}{48}+\tfrac{(55+100\sqrt{3}+36\sqrt{11}+17\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(100\sqrt{3}+17\sqrt{33}-55-36\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\hat{\hat{\tau}}_{\scriptsize11,5}&={\scriptsize-\tfrac{583+30\sqrt{11}}{1056}-\tfrac{(25+50\sqrt{3}+18\sqrt{11}+7\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(50\sqrt{3}+7\sqrt{33}-25-18\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\

\hat{\hat{\omega}}_{\scriptsize11,1}&={\scriptsize-\tfrac{30-\sqrt{11}}{6}-\tfrac{(11\sqrt{3}+5\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}-\tfrac{(11\sqrt{3}-5\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\hat{\hat{\omega}}_{\scriptsize11,2}&={\scriptsize\tfrac{60-\sqrt{11}}{6}-\tfrac{(33+110\sqrt{3}+46\sqrt{11}-\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(33-110\sqrt{3}+46\sqrt{11}+\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\hat{\hat{\omega}}_{\scriptsize11,3}&={\scriptsize-\tfrac{5(6+\sqrt{11})}{3}-\tfrac{(66+253\sqrt{3}+113\sqrt{11}+30\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(253\sqrt{3}+30\sqrt{33}-66-113\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\hat{\hat{\omega}}_{\scriptsize11,4}&={\scriptsize-\tfrac{37(2\sqrt{11}-5)}{48}-\tfrac{(264\sqrt{3}+112\sqrt{11}-253-71\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(112\sqrt{11}+71\sqrt{33}-253-264\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\hat{\hat{\omega}}_{\scriptsize11,5}&={\scriptsize-\tfrac{30+53\sqrt{11}}{96}-\tfrac{(198+77\sqrt{3}+25\sqrt{11}+50\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(77\sqrt{3}+50\sqrt{33}-198-25\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\
\end{split}

\end{gather*}

1. Sqrt[11] (-((583 - 30 Sqrt[11])/
2.     1056) + ((50 Sqrt[3] + 18 Sqrt[11] - 25 -
3.       7 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
4.    192 + ((25 + 50 Sqrt[3] - 18 Sqrt[11] - 7 Sqrt[33]) (17 +
5.       3 Sqrt[33])^(1/3))/192)/((44 + 33 Sqrt[11])/
6.    96 + ((220 + 33 Sqrt[3] + 45 Sqrt[11] + 20 Sqrt[33]) (3 Sqrt[33] -
7.       17)^(1/3))/
8.    192 - ((220 - 33 Sqrt[3] + 45 Sqrt[11] - 20 Sqrt[33]) (17 +
9.       3 Sqrt[33])^(1/3))/192) // N
10. (k*M)/l /. {
11.    k -> ((3 Sqrt[2] + Sqrt[22])/
12.      12 - ((Sqrt[2] + Sqrt[6]) (Sqrt[3] + Sqrt[11]) (3 Sqrt[33] -
13.         17)^(1/3))/
14.      24 - ((Sqrt[6] - Sqrt[2]) (Sqrt[11] - Sqrt[3]) (17 +
15.         3 Sqrt[33])^(1/3))/24)/1,
16.    l -> ((Sqrt[22] - 3 Sqrt[2])/
17.      12 + ((Sqrt[6] - Sqrt[2]) (Sqrt[3] + Sqrt[11]) (3 Sqrt[33] -
18.         17)^(1/3))/
19.      24 + ((Sqrt[2] + Sqrt[6]) (Sqrt[11] - Sqrt[3]) (17 +
20.         3 Sqrt[33])^(1/3))/24)/1, M -> 1/Sqrt[11]} // N
21. N[{((3 Sqrt[2] + Sqrt[22])/
22.     12 - ((Sqrt[2] + Sqrt[6]) (Sqrt[3] + Sqrt[11]) (3 Sqrt[33] - 17)^(
23.      1/3))/24 - ((Sqrt[6] - Sqrt[2]) (Sqrt[11] - Sqrt[3]) (17 +
24.        3 Sqrt[33])^(1/3))/24)/1,
25.    ((Sqrt[22] - 3 Sqrt[2])/
26.     12 + ((Sqrt[6] - Sqrt[2]) (Sqrt[3] + Sqrt[11]) (3 Sqrt[33] - 17)^(
27.      1/3))/24 + ((Sqrt[2] + Sqrt[6]) (Sqrt[11] - Sqrt[3]) (17 +
28.        3 Sqrt[33])^(1/3))/24)/1,
29.    Sqrt[ModularLambda[I Sqrt[11]]], Sqrt[ModularLambda[I/Sqrt[11]]]},
30.   20] // Column

复制代码

1. \tau_{1}=-\frac{5}{2}+\frac{(7\sqrt{3}+3\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\frac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}
2. \tau_{2}=4-\frac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
3. \tau_{3}=-\frac{14-\sqrt{11}}{4}+\frac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}+\frac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33})(17+3\sqrt{33})^{1/3}}{48}
4. \tau_{4}=\frac{101-12\sqrt{11}}{48}-\frac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}-\frac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33})^{1/3}}{96}
5. \tau_{5}=-\frac{583-30\sqrt{11}}{1056}+\frac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}+\frac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33})(17+3\sqrt{33})^{1/3}}{192}
6. \omega_{a1}=-\frac{30+\sqrt{11}}{6}+\frac{(11\sqrt{3}+5\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}+\frac{(11\sqrt{3}-5\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}
7. \omega_{a2}=\frac{60-\sqrt{11}}{6}-\frac{(33+110\sqrt{3}+46\sqrt{11}-\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(33-110\sqrt{3}+46\sqrt{11}+\sqrt{33})(17+3\sqrt{33})^{1/3}}{24}
8. \omega_{a3}=-\frac{5(6-\sqrt{11})}{3}+\frac{(253\sqrt{3}+113\sqrt{11}-66-30\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}-\frac{(113\sqrt{11}+30\sqrt{33}-66-253\sqrt{3})(17+3\sqrt{33})^{1/3}}{48}
9. \omega_{a4}=-\frac{37(2\sqrt{11}-5)}{48}-\frac{(264\sqrt{3}+112\sqrt{11}-253-71\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}+\frac{(112\sqrt{11}+71\sqrt{33}-253-264\sqrt{3})(17+3\sqrt{33})^{1/3}}{96}
10. \omega_{a5}=\frac{53\sqrt{11}-30}{96}-\frac{(198+50\sqrt{33}-77\sqrt{3}-25\sqrt{11})(3\sqrt{33}-17)^{1/3}}{192}-\frac{(25\sqrt{11}+50\sqrt{33}-198-77\sqrt{3})(17+3\sqrt{33})^{1/3}}{192}
11. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt
12. \frac{1}{\sqrt{11}}\int_{0}^{f\left(x,1\right)}\frac{1}{\sqrt{(1+t^{2})(1+k^{2}t^{2})}}dt
13. k=\frac{3\sqrt{2}+\sqrt{22}}{12}-\frac{(\sqrt{2}+\sqrt{6})(\sqrt{3}+\sqrt{11})(3\sqrt{33}-17)^{1/3}}{24}-\frac{(\sqrt{6}-\sqrt{2})(\sqrt{11}-\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
14. f\left(x,v\right)=\frac{\sqrt{11}x(1+\tau_{1}x^{2}+\tau_{2}x^{4}+\tau_{3}x^{6}+\tau_{4}x^{8}+\tau_{5}x^{10})}{\sqrt{1-x^{2}}(1+\omega_{a1}x^{2}+\omega_{a2}x^{4}+\omega_{a3}x^{6}+\omega_{a4}x^{8}+\omega_{a5}x^{10})}

复制代码

1. NIntegrate[
2. 1/Sqrt[(1 - t^2) (1 - m^2 t^2)] /. {m -> Sqrt[
3.     ModularLambda[Sqrt[11] I]]}, {t, 0, y /. {y -> 232/1000}},
4. WorkingPrecision -> 30]
5. NIntegrate[
6. 1/Sqrt[11]/
7.   Sqrt[(1 + t^2) (1 + m^2 t^2)] /. {m -> Sqrt[
8.     ModularLambda[Sqrt[11] I]]}, {t, 0, (Sqrt[11] x (
9.        1
10.         + (-(5/2) + ((7 Sqrt[3] + 3 Sqrt[11]) (3 Sqrt[33] - 17)^(
11.             1/3))/6 + ((7 Sqrt[3] - 3 Sqrt[11]) (17 + 3 Sqrt[33])^(
12.             1/3))/6 ) x^2
13.         + (4 - ( (56 Sqrt[3] + 24 Sqrt[11] - 15 -
14.               5 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
15.            24 - ((15 + 56 Sqrt[3] - 24 Sqrt[11] - 5 Sqrt[33]) (17 +
16.               3 Sqrt[33])^(1/3))/24) x^4
17.         + (-((14 - Sqrt[11])/
18.             4) + ((106 Sqrt[3] + 42 Sqrt[11] - 45 -
19.               15 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
20.            48  + ( (45 + 106 Sqrt[3] - 42 Sqrt[11] -
21.               15 Sqrt[33]) (17 + 3 Sqrt[33])^(1/3))/48) x^6
22.         + ((101 - 12 Sqrt[11])/
23.            48 - ((-55 + 100 Sqrt[3] + 36 Sqrt[11] -
24.               17 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
25.            96  - ( (55 + 100 Sqrt[3] - 36 Sqrt[11] -
26.               17 Sqrt[33]) (17 + 3 Sqrt[33])^(1/3))/96) x^8
27.         + (-((583 - 30 Sqrt[11])/
28.             1056) + ((50 Sqrt[3] + 18 Sqrt[11] - 25 -
29.               7 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
30.            192 + ((25 + 50 Sqrt[3] - 18 Sqrt[11] - 7 Sqrt[33]) (17 +
31.               3 Sqrt[33])^(1/3))/192) x^10)
32.      )/(Sqrt[(1 - x) (1 + x)] (1
33.         - ((30 + Sqrt[11])/
34.            6 - ((11 Sqrt[3] + 5 Sqrt[11]) (3 Sqrt[33] - 17)^(1/3))/
35.            6 - ((11 Sqrt[3] - 5 Sqrt[11]) (17 + 3 Sqrt[33])^(1/3))/
36.            6) x^2
37.         + ((60 - Sqrt[11])/
38.            6 - ((33 + 110 Sqrt[3] + 46 Sqrt[11] - Sqrt[
39.               33]) (3 Sqrt[33] - 17)^(1/3))/
40.            24 + ((33 - 110 Sqrt[3] + 46 Sqrt[11] + Sqrt[33]) (17 +
41.               3 Sqrt[33])^(1/3))/24) x^4
42.         - ((5 (6 - Sqrt[11]))/
43.            3 - ((253 Sqrt[3] + 113 Sqrt[11] - 66 -
44.               30 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
45.            48 + ((113 Sqrt[11] + 30 Sqrt[33] - 66 -
46.               253 Sqrt[3]) (17 + 3 Sqrt[33])^(1/3))/48  ) x^6
47.         + (-((37 (2 Sqrt[11] - 5))/
48.             48) - ( (264 Sqrt[3] + 112 Sqrt[11] - 253 -
49.               71 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
50.            96  + ( (112 Sqrt[11] + 71 Sqrt[33] - 253 -
51.               264 Sqrt[3]) ((17 + 3 Sqrt[33])^(1/3)) )/96 ) x^8
52.         - (-((53 Sqrt[11] - 30)/
53.             96) + ( (198 + 50 Sqrt[33] - 77 Sqrt[3] -
54.               25 Sqrt[11]) (3 Sqrt[33] - 17)^(1/3))/
55.            192 + ((25 Sqrt[11] + 50 Sqrt[33] - 198 -
56.               77 Sqrt[3]) (17 + 3 Sqrt[33])^(1/3))/192  ) x^10)
57.      ) /. {x -> 232/1000}}, WorkingPrecision -> 30]

复制代码

1. \tau_{b1}=-\frac{5}{2}-\frac{(7\sqrt{3}+3\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}-\frac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}
2. \tau_{b2}=4+\frac{(15+56\sqrt{3}+24\sqrt{11}+5\sqrt{33})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(56\sqrt{3}+5\sqrt{33}-15-24\sqrt{11})(17+3\sqrt{33})^{1/3}}{24}
3. \tau_{b3}=-\frac{14+\sqrt{11}}{4}-\frac{(45+106\sqrt{3}+42\sqrt{11}+15\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}-\frac{(106\sqrt{3}+15\sqrt{33}-45-42\sqrt{11})(17+3\sqrt{33})^{1/3}}{48}
4. \tau_{b4}=\frac{101+12\sqrt{11}}{48}+\frac{(55+100\sqrt{3}+36\sqrt{11}+17\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}+\frac{(100\sqrt{3}+17\sqrt{33}-55-36\sqrt{11})(17+3\sqrt{33})^{1/3}}{96}
5. \tau_{b5}=-\frac{583+30\sqrt{11}}{1056}-\frac{(25+50\sqrt{3}+18\sqrt{11}+7\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}-\frac{(50\sqrt{3}+7\sqrt{33}-25-18\sqrt{11})(17+3\sqrt{33})^{1/3}}{192}
6. \omega_{b1}=-\frac{30-\sqrt{11}}{6}-\frac{(11\sqrt{3}+5\sqrt{11})(3\sqrt{33}-17)^{1/3}}{6}-\frac{(11\sqrt{3}-5\sqrt{11})(17+3\sqrt{33})^{1/3}}{6}
7. \omega_{b2}=\frac{60+\sqrt{11}}{6}+\frac{(110\sqrt{3}+46\sqrt{11}+\sqrt{33}-33)(3\sqrt{33}-17)^{1/3}}{24}+\frac{(33+110\sqrt{3}-46\sqrt{11}+\sqrt{33})(17+3\sqrt{33})^{1/3}}{24}
8. \omega_{b3}=-\frac{5(6+\sqrt{11})}{3}-\frac{(66+253\sqrt{3}+113\sqrt{11}+30\sqrt{33})(3\sqrt{33}-17)^{1/3}}{48}-\frac{(253\sqrt{3}+30\sqrt{33}-66-113\sqrt{11})(17+3\sqrt{33})^{1/3}}{48}
9. \omega_{b4}=\frac{37(5+2\sqrt{11})}{48}+\frac{(253+264\sqrt{3}+112\sqrt{11}+71\sqrt{33})(3\sqrt{33}-17)^{1/3}}{96}+\frac{(264\sqrt{3}+71\sqrt{33}-253-112\sqrt{11})(17+3\sqrt{33})^{1/3}}{96}
10. \omega_{b5}=-\frac{30+53\sqrt{11}}{96}-\frac{(198+77\sqrt{3}+25\sqrt{11}+50\sqrt{33})(3\sqrt{33}-17)^{1/3}}{192}-\frac{(77\sqrt{3}+50\sqrt{33}-198-25\sqrt{11})(17+3\sqrt{33})^{1/3}}{192}
11. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-l^{2}t^{2})}}dt
12. \frac{1}{\sqrt{11}}\int_{0}^{f\left(x,1\right)}\frac{1}{\sqrt{(1+t^{2})(1+l^{2}t^{2})}}dt
13. l=\frac{\sqrt{22}-3\sqrt{2}}{12}+\frac{(\sqrt{6}-\sqrt{2})(\sqrt{3}+\sqrt{11})(3\sqrt{33}-17)^{1/3}}{24}+\frac{(\sqrt{2}+\sqrt{6})(\sqrt{11}-\sqrt{3})(17+3\sqrt{33})^{1/3}}{24}
14. f\left(x,v\right)=\frac{\sqrt{11}x(1+\tau_{b1}x^{2}+\tau_{b2}x^{4}+\tau_{b3}x^{6}+\tau_{b4}x^{8}+\tau_{b5}x^{10})}{\sqrt{1-x^{2}}(1+\omega_{b1}x^{2}+\omega_{b2}x^{4}+\omega_{b3}x^{6}+\omega_{b4}x^{8}+\omega_{b5}x^{10})}

复制代码

1. NIntegrate[
2. 1/Sqrt[(1 - t^2) (1 - (1 - m^2) t^2)] /. {m -> Sqrt[
3.     ModularLambda[Sqrt[11] I]]}, {t, 0, y /. {y -> 232/1000}},
4. WorkingPrecision -> 30]
5. NIntegrate[
6. 1/Sqrt[11]/(
7.   I Sqrt[(1 - t^2) (1 - (1 - m^2) t^2)]) /. {m -> Sqrt[
8.     ModularLambda[Sqrt[11] I]]}, {t, 0, I*Y Sqrt[11] (1
9.        + (-(5/2) - ((7 Sqrt[3] + 3 Sqrt[11]) (3 Sqrt[33] - 17)^(1/3))/
10.           6 - ((7 Sqrt[3] - 3 Sqrt[11]) (17 + 3 Sqrt[33])^(1/3))/
11.           6 ) Y^2
12.        + (4 + ( (15 + 56 Sqrt[3] + 24 Sqrt[11] +
13.              5 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
14.           24 + ((56 Sqrt[3] + 5 Sqrt[33] - 15 - 24 Sqrt[11]) (17 +
15.              3 Sqrt[33])^(1/3))/24   ) Y^4
16.        + (-((14 + Sqrt[11])/
17.            4) - ( (45 + 106 Sqrt[3] + 42 Sqrt[11] +
18.              15 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
19.           48 - ( (106 Sqrt[3] + 15 Sqrt[33] - 45 - 42 Sqrt[11]) (17 +
20.              3 Sqrt[33])^(1/3))/48 ) Y^6
21.        + ((101 + 12 Sqrt[11])/
22.           48 + ((55 + 100 Sqrt[3] + 36 Sqrt[11] +
23.              17 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
24.           96 + ( (100 Sqrt[3] + 17 Sqrt[33] - 55 - 36 Sqrt[11]) (17 +
25.              3 Sqrt[33])^(1/3))/96) Y^8
26.        + (-((583 + 30 Sqrt[11])/
27.            1056) - ( (25 + 50 Sqrt[3] + 18 Sqrt[11] +
28.              7 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
29.           192 - ( (50 Sqrt[3] + 7 Sqrt[33] - 25 - 18 Sqrt[11]) (17 +
30.              3 Sqrt[33])^(1/3))/192 ) Y^10)/(Sqrt[1 - Y^2] (
31.         1
32.          + (-((30 - Sqrt[11])/
33.              6)  - ((11 Sqrt[3] + 5 Sqrt[11]) (3 Sqrt[33] - 17)^(
34.              1/3))/6  - ((11 Sqrt[3] - 5 Sqrt[11]) (17 + 3 Sqrt[33])^(
35.              1/3))/6 ) Y^2
36.          + ((60 + Sqrt[11])/
37.             6 + ( (33 + 110 Sqrt[3] - 46 Sqrt[11] + Sqrt[33]) ((17 +
38.                3 Sqrt[33])^(1/3)) )/
39.             24 + ((110 Sqrt[3] + 46 Sqrt[11] + Sqrt[33] -
40.                33) (3 Sqrt[33] - 17)^(1/3))/24  ) Y^4
41.          + (-((5 (6 + Sqrt[11]))/
42.              3)  - ( (66 + 253 Sqrt[3] + 113 Sqrt[11] +
43.                30 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
44.             48  - ( (253 Sqrt[3] + 30 Sqrt[33] - 66 -
45.                113 Sqrt[11]) (17 + 3 Sqrt[33])^(1/3))/48  ) Y^6
46.          + ((37 (5 + 2 Sqrt[11]))/
47.             48  + ( (253 + 264 Sqrt[3] + 112 Sqrt[11] +
48.                71 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
49.             96 + ((264 Sqrt[3] + 71 Sqrt[33] - 253 -
50.                112 Sqrt[11]) ((17 + 3 Sqrt[33])^(1/3))  )/96  ) Y^8
51.          + (-((30 + 53 Sqrt[11])/
52.              96) - ((198 + 77 Sqrt[3] + 25 Sqrt[11] +
53.                50 Sqrt[33]) (3 Sqrt[33] - 17)^(1/3))/
54.             192 - ((17 + 3 Sqrt[33])^(
55.              1/3) (77 Sqrt[3] + 50 Sqrt[33] - 198 - 25 Sqrt[11]))/
56.             192  ) Y^10)) /. Y -> 232/1000
57.   }, WorkingPrecision -> 30]

复制代码

青青子衿

青青子衿 发表于 2023-9-29 23:23
\begin{align*}
j(\tau)&=\frac{\left(\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^{12}+250 \left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^6+3125\right)^3}{\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^{30}}\\
j(5\tau)&=\frac{\left(\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^{12}+10 \left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^6+5\right)^3}{\left(\frac{\eta (\tau)}{\eta (5\tau)}\right)^6}
\end{align*}

\begin{align*}
j(\tau)&=\frac{(t+27) (t+3)^3}{t}\\
j(3\tau)&=\frac{(t+27) (t+243)^3}{t^3}\\
t&=3^6\frac{\eta^{12}(3\tau)}{\eta ^{12}(\tau)}
\end{align*}
\begin{align*}
j(\tau)&=\frac{(t^2+10 t+5)^3}{t}\\
j(5\tau)&=\frac{(t^2+250 t+3125)^3}{t^5}\\
t&=5^3\frac{\eta^{6}(5\tau)}{\eta ^{6}(\tau)}
\end{align*}
\begin{align*}
j(\tau)&=\frac{(t+6)^3(t^3+18t^2+84t+24)^3}{t(t+8)^3(t+9)^2}\\
j(6\tau)&=\frac{\left(t+12\right)^3\left(t^3+252t^2+3888t+15552\right)^3}{t^6(t+8)^2(t+9)^3}\\
t&=2^3 3^2 \frac{\eta^5(6\tau)\eta(2\tau)}{\eta(3\tau)\eta^5(\tau)}
\end{align*}

Parametrization of the modular curve X0(N) for N from 2 to 37
math.fsu.edu/~hoeij/files/X0N/
On Rationally Parametrized Modular Equations
Robert S. Maier
arxiv.org/abs/math/0611041


\begin{align*}
k\Big(\tau=\tfrac{1+2 i\sqrt{2}}{3} \Big)=\frac{(\sqrt{2}-1)(1+i)(2+\sqrt{2}-i\sqrt{\scriptsize1+\sqrt{2}})}{4}(2+2\sqrt{2})^{1/4}
\end{align*}

青青子衿

青青子衿 发表于 2024-9-2 08:55
\begin{gather*}
I_{11}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}
=\frac{1}{i\sqrt{11}}\int_{0}^{\hat{y}_{\scriptsize11}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{11}t^{2})}}\\
\\
\hat{y}_{\scriptsize11}=\tfrac{i\sqrt{\scriptsize11}\,x(1+\tau_{\scriptsize11,1}x^2+\tau_{\scriptsize11,2}x^4+\tau_{\scriptsize11,3}x^6+\tau_{\scriptsize11,4}x^8+\tau_{\scriptsize11,5}x^{10})}{\sqrt{1-x^2}(1+\hat\omega_{\scriptsize11,1}x^2+\hat\omega_{\scriptsize11,2}x^4+\hat\omega_{\scriptsize11,3}x^6+\hat\omega_{\scriptsize11,4}x^8+\hat\omega_{\scriptsize11,5}x^{10})}\\
\\
\begin{split}
U_{\scriptsize11}&={\scriptsize\left(\tfrac{3\sqrt{2}+\sqrt{22}}{12}
-\tfrac{(\sqrt{2}+\sqrt{6}\>\!)(\sqrt{3}+\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}-\tfrac{(\sqrt{6}-\sqrt{2}\>\!)(\sqrt{11}-\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}\right)^2}

\end{split}\\
\\
\qquad\begin{split}
\tau_{\scriptsize11,1}&={\scriptsize-\tfrac{5}{2}+\tfrac{(7\sqrt{3}+3\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(7\sqrt{3}-3\sqrt{11})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\tau_{\scriptsize11,2}&={\scriptsize4-\tfrac{(56\sqrt{3}+24\sqrt{11}-15-5\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(24\sqrt{11}+5\sqrt{33}-15-56\sqrt{3})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\tau_{\scriptsize11,3}&={\scriptsize-\tfrac{14-\sqrt{11}}{4}+\tfrac{(106\sqrt{3}+42\sqrt{11}-45-15\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}+\tfrac{(45+106\sqrt{3}-42\sqrt{11}-15\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\tau_{\scriptsize11,4}&={\scriptsize\tfrac{101-12\sqrt{11}}{48}-\tfrac{(100\sqrt{3}+36\sqrt{11}-55-17\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}-\tfrac{(55+100\sqrt{3}-36\sqrt{11}-17\sqrt{33})(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\tau_{\scriptsize11,5}&={\scriptsize-\tfrac{583-30\sqrt{11}}{1056}+\tfrac{(50\sqrt{3}+18\sqrt{11}-25-7\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}+\tfrac{(25+50\sqrt{3}-18\sqrt{11}-7\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\

\hat\omega_{\scriptsize11,1}&={\scriptsize-\tfrac{30+\sqrt{11}}{6}+\tfrac{(11\sqrt{3}+5\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{6}+\tfrac{(11\sqrt{3}-5\sqrt{11}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{6}}\\

\hat\omega_{\scriptsize11,2}&={\scriptsize\tfrac{60-\sqrt{11}}{6}-\tfrac{(33+110\sqrt{3}+46\sqrt{11}-\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{24}+\tfrac{(33-110\sqrt{3}+46\sqrt{11}+\sqrt{33}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{24}}\\

\hat\omega_{\scriptsize11,3}&={\scriptsize-\tfrac{5(6-\sqrt{11}\>\>\!\!)}{3}+\tfrac{(253\sqrt{3}+113\sqrt{11}-66-30\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{48}-\tfrac{(113\sqrt{11}+30\sqrt{33}-66-253\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{48}}\\

\hat\omega_{\scriptsize11,4}&={\scriptsize-\tfrac{37(2\sqrt{11}-5)}{48}-\tfrac{(264\sqrt{3}+112\sqrt{11}-253-71\sqrt{33}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{96}+\tfrac{(112\sqrt{11}+71\sqrt{33}-253-264\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{96}}\\

\hat\omega_{\scriptsize11,5}&={\scriptsize\tfrac{53\sqrt{11}-30}{96}-\tfrac{(198+50\sqrt{33}-77\sqrt{3}-25\sqrt{11}\>\>\!\!)(3\sqrt{33}-17)^{1/3}}{192}-\tfrac{(25\sqrt{11}+50\sqrt{33}-198-77\sqrt{3}\>\>\!\!)(17+3\sqrt{33}\>\>\!\!)^{1/3}}{192}}\\
\end{split}

\end{gather*}

\begin{gather*}
I_{8}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{8}t^{2})}}
=\frac{1}{2i\sqrt{2}}\int_{0}^{\hat{y}_{\scriptsize8}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{8}t^{2})}}\\
\\
\hat{y}_{\scriptsize8}=\left(\tfrac{1+\tau_{\scriptsize8,1}x^{2}+\tau_{\scriptsize8,2}x^{4}+\tau_{\scriptsize8,3}x^{6}}{1+\hat\omega_{\scriptsize8,1}x^{2}+\hat\omega_{\scriptsize8,2}x^{4}+\hat\omega_{\scriptsize8,3}x^{6}}\right)\tfrac{2i\sqrt{2}x}{\sqrt{(1-x^{2})(1-U_{8}x^{2})}}\\
\\
\begin{split}
U_{\scriptsize8}&={\scriptsize\left(5+4\sqrt{2}-2\sqrt{\Tiny14+10\sqrt{2}}\,\right)^2}

\end{split}\\
\\
\qquad\begin{split}
\tau_{\scriptsize8,1}&={\scriptsize67+48\sqrt{2}-2\sqrt{\Tiny2254+1594\sqrt{2}}}\\

\tau_{\scriptsize8,2}&={\scriptsize1431+1012\sqrt{2}-4\sqrt{\Tiny255998+181018\sqrt{2}}}\\

\tau_{\scriptsize8,3}&={\scriptsize2405+1700\sqrt{2}-2\sqrt{\Tiny2891006+2044250\sqrt{2}}}\\



\hat\omega_{\scriptsize8,1}&={\scriptsize-(47+32\sqrt{2}-2\sqrt{\Tiny1054+746\sqrt{2}}\,)}\\

\hat\omega_{\scriptsize8,2}&={\scriptsize979+692\sqrt{2}-4\sqrt{\Tiny119758+84682\sqrt{2}}}\\

\hat\omega_{\scriptsize8,3}&={\scriptsize-(2405+1700\sqrt{2}-2\sqrt{\Tiny2891006+2044250\sqrt{2}}\,)}\\


\end{split}

\end{gather*}


\begin{align*}
\int_{0}^{x}\frac{dt}{\sqrt{(1-t^{2})(1-({\scriptsize\sqrt{2}-1})^{2}t^{2})}}=
\frac{1}{i\sqrt{2}}\int_{0}^{\frac{i\sqrt{2}x}{\sqrt{(1-x^{2})(1-({\scriptsize\sqrt{2}-1})^{2}x^{2})}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-({\scriptsize\sqrt{2}-1})^{2}t^{2})}}
\end{align*}



\begin{gather*}
\begin{split}
\int_{0}^{x}\frac{dt}{\sqrt{(1-t^{2})(1-({\scriptsize\sqrt{2}-1})^{2}t^{2})}}&=
\frac{1}{-1-2i\sqrt{2}}\int_{0}^{y}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-({\scriptsize\sqrt{2}-1})^{2}t^{2})}}\\
&=\frac{1}{(1-i\sqrt{2})^2}\int_{0}^{w{\circ}w}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-({\scriptsize\sqrt{2}-1})^{2}t^{2})}}\\
\end{split}\\

\\
y=\tfrac{(-1-2i\sqrt{2})x\left(1-\frac{2(\sqrt{2}-1)(2\sqrt{2}+i)}{3}x^{2}+2(3\sqrt{2}-4)ix^{4}-2i(5\sqrt{2}-7)x^{6}-\frac{17-12\sqrt{2}-2i(17\sqrt{2}-24)}{9}x^{8}\right)}{1-2(\sqrt{2}-1)(2\sqrt{2}-i)x^{2}+2(3\sqrt{2}-4)(2\sqrt{2}-i)x^{4}+6i(5\sqrt{2}-7)x^{6}-\frac{(17\sqrt{2}-24)(\sqrt{2}+4i)}{2}x^{8}}\\

w=\tfrac{(1-i\sqrt{2})x(1-\frac{(1+i)(3-i-2\sqrt{2})}{6}x^{2})}{1-\frac{(1-i)(3+i-2\sqrt{2})}{2}x^{2}}
\end{gather*}


\begin{gather*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{8}t^{2})}}

=\frac{1}{1+2i\sqrt{2}}\int_{0}^{\hat{y}_{\scriptsize}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-U_{8}t^{2})}}\\
\\
\hat{y}_{\scriptsize}=\tfrac{(1+2 i \sqrt{2})x(1+\tau_{\scriptsize1}x^{2}+\tau_{\scriptsize2}x^{4}+\tau_{\scriptsize3}x^{6}+\tau_{\scriptsize4}x^{8})}{1+\hat\omega_{\scriptsize1}x^{2}+\hat\omega_{\scriptsize2}x^{4}+\hat\omega_{\scriptsize3}x^{6}+\hat\omega_{\scriptsize4}x^{8}}\\
\\
\begin{split}
U_{\scriptsize8}&={\scriptsize\left(5+4\sqrt{2}-2\sqrt{\Tiny14+10\sqrt{2}}\,\right)^2}

\end{split}\\
\\
\qquad\begin{split}
\tau_{\scriptsize1}&={\small\tfrac{2(2-3i)(75+2i+52\sqrt{2})(2+3\sqrt{2}-4\sqrt{1+\sqrt{2}})}{39}}\\

\tau_{\scriptsize2}&={\small-\tfrac{(137+106i)(42478-14i+30005\sqrt{2})(46+29\sqrt{2}-56\sqrt{1+\sqrt{2}})}{210035}}\\

\tau_{\scriptsize3}&={\small\tfrac{2i(1321+934\sqrt{2})(1670+265\sqrt{2}-1316\sqrt{1+\sqrt{2}})}{329}}\\

\tau_{\scriptsize4}&={\small\tfrac{(2909-8228i)(107708634+868i+76162265\sqrt{2})(4707\sqrt{2}-1262-3472\sqrt{1+\sqrt{2}})}{297489807090}}\\

\hat\omega_{\scriptsize1}&={\small-\tfrac{(3-i)(27-i+20\sqrt{2})(2+3\sqrt{2}-4\sqrt{1+\sqrt{2}})}{5}}\\

\hat\omega_{\scriptsize2}&={\small\tfrac{(163-494i)(382302-154i+270605\sqrt{2})(46+29\sqrt{2}-56\sqrt{1+\sqrt{2}})}{1894235}}\\

\hat\omega_{\scriptsize3}&={\small\tfrac{2(3189+854i)(30827190-1645i+21798074\sqrt{2})(1670+265\sqrt{2}-1316\sqrt{1+\sqrt{2}})}{3585783173}}\\

\hat\omega_{\scriptsize4}&={\small\tfrac{(2909+8228i)(107708634-868i+76162265\sqrt{2})(4707\sqrt{2}-1262-3472\sqrt{1+\sqrt{2}})}{33054423010}}\\
\end{split}

\end{gather*}


\begin{gather*}
{\large{\int}}_{0 }^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-(3-2\sqrt{2})^{2}t^{2})}}=\frac{1}{2i}{\large{\int}}_{0 }^{y}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-(3-2\sqrt{2})^{2}t^{2})}}\\
\\
y=\tfrac{2ix(1+(3-2\sqrt{2})x^{2})}{1-2(10-7\sqrt{2})x^{2}+(99-70\sqrt{2})x^{4}}\Big(\tfrac{1-(3-2\sqrt{2})^{2}x^{2}}{1-x^{2}}\Big)^{1/2}
\end{gather*}


\begin{gather*}
{\large{\int}}_{0 }^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\scriptsize\raise{0.5pt}\frac{(2-\sqrt{2})2^{1/4}}{4}i})^{2}t^{2})}}=\frac{1}{1+2i}{\large{\int}}_{0 }^{y}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\scriptsize\raise{0.5pt}\frac{(2-\sqrt{2})2^{1/4}}{4}i})^{2}t^{2})}}\\
\\
y=\tfrac{(1+2i)x}{\sqrt{1-x^{2}}}\Big(\tfrac{1+\frac{(3-i)(5\sqrt{2}-6-2i)}{20}x^{2}-\frac{(3-i)(5\sqrt{2}-7)}{40}x^{4}}{1+\frac{(2-i)(5\sqrt{2}-8+i)}{10}x^{2}-\frac{(1-i)(5\sqrt{2}-7)}{8}x^{4}}\Big)
\end{gather*}


\begin{gather*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\lambda}t^{2})}}

=\frac{1}{1+2i\sqrt{2}}\int_{0}^{y}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\lambda}t^{2})}}\\
\\
y_{\scriptsize}=\tfrac{(1+2 i \sqrt{2})x}{\sqrt{1-x^2}}\left(\tfrac{1+u_{\scriptsize1}x^{2}+u_{\scriptsize2}x^{4}+u_{\scriptsize3}x^{6}+u_{\scriptsize4}x^{8}}{1+v_{\scriptsize1}x^{2}+v_{\scriptsize2}x^{4}+v_{\scriptsize3}x^{6}+v_{\scriptsize4}x^{8}}\right)\\
\\
\begin{split}
\lambda&={\scriptsize\left(i\cdot\frac{\sqrt{2}-2(\sqrt{2}-1)\sqrt{1+\sqrt{2}}}{4}(2+2\sqrt{2})^{1/4}\right)^2}

\end{split}\\
\\
\qquad\begin{split}
u_{\scriptsize1}&={\small\tfrac{-12+(8-i)\sqrt{1+\sqrt{2}}+(1-i)\sqrt{2+2\sqrt{2}}}{6}}\\

u_{\scriptsize2}&={\small-\tfrac{-19+5i-(7-2i)\sqrt{2}+2(8-i)\sqrt{1+\sqrt{2}}+2(1-i)\sqrt{2+2\sqrt{2}}}{8}}\\

u_{\scriptsize3}&={\small-\tfrac{22-10i+2(7-2i)\sqrt{2}-(17-3i)\sqrt{1+\sqrt{2}}-(7-5i)\sqrt{2+2\sqrt{2}}}{16}}\\

u_{\scriptsize4}&={\small\tfrac{216-128i+(131-45i)\sqrt{2}-6(19-5i)\sqrt{1+\sqrt{2}}-6(17-11i)\sqrt{2+2\sqrt{2}}}{576}}\\

v_{\scriptsize1}&={\small-\tfrac{7-2i\sqrt{2}-4\sqrt{1+\sqrt{2}}+i\sqrt{2+2\sqrt{2}}}{2}}\\

v_{\scriptsize2}&={\small-\tfrac{-37+15i-(12-13i)\sqrt{2}+2(16-5i)\sqrt{1+\sqrt{2}}+2(1-4i)\sqrt{2+2\sqrt{2}}}{8}}\\

v_{\scriptsize3}&={\small\tfrac{-47+33i-2(15-14i)\sqrt{2}+2(21-17i)\sqrt{1+\sqrt{2}}+(11-9i)\sqrt{2+2\sqrt{2}}}{16}}\\

v_{\scriptsize4}&={\small\tfrac{(1-i)(44+43\sqrt{2}-42\sqrt{1+\sqrt{2}}-18\sqrt{2+2\sqrt{2}}\>\!)}{64}}\\
\end{split}

\end{gather*}

青青子衿

\begin{align*}
I_0&=K\left(\tau=\tfrac{1+i\sqrt{17}}{3}\right)\\

&=
K\left(k={\small(\sqrt{17}-4)(5i-2\sqrt{\scriptsize2+2\sqrt{17}})+\tfrac{\sqrt{2}(1-i)(21-5\sqrt{17})(9+\sqrt{17}+2i\sqrt{2+2\sqrt{17}})}{16}(2+2\sqrt{17})^{1/4}}\right)\\

&=\left(\small\tfrac{2(-3+2i)\sqrt{2}-2\sqrt{34}+(7+i)\sqrt{1+\sqrt{17}}+(1-i)\sqrt{17+17\sqrt{17}}}{8}\right.\\
&\qquad\qquad\left.\small\tfrac{-4(3+4i)-4\sqrt{17}+(7-2i)\sqrt{2+2\sqrt{17}}+(1+2i)\sqrt{34+34\sqrt{17}}}{32}(2+2\sqrt{17})^{1/4}\right)\\
&\qquad\qquad\qquad\,K\left(k={\small\tfrac{(3+\sqrt{17})(2\sqrt{2}-\sqrt{1+\sqrt{17}})}{8}-\tfrac{(7+\sqrt{17})(4-\sqrt{2+2\sqrt{17}})}{32}(2+2\sqrt{17})^{1/4}}\right)\\

&=\left(\small\tfrac{2(-3+2i)\sqrt{2}-2\sqrt{34}+(7+i)\sqrt{1+\sqrt{17}}+(1-i)\sqrt{17+17\sqrt{17}}}{8}\right.\\
&\qquad\qquad\left.\small\tfrac{-4(3+4i)-4\sqrt{17}+(7-2i)\sqrt{2+2\sqrt{17}}+(1+2i)\sqrt{34+34\sqrt{17}}}{32}(2+2\sqrt{17})^{1/4}\right)\\
&\qquad\qquad\qquad\,
\small\tfrac{(6+2\sqrt{17}+5\sqrt{4+\sqrt{17}})^{1/4}}{2^{81/34}17^{9/16}}
\left(\tfrac{\tan(\frac{\pi}{68})\tan(\frac{9\pi}{68})\tan(\frac{13\pi}{68})}{\tan(\frac{15\pi}{68})}\right)^{1/4}
\left(\tfrac{\Gamma^{2}(\frac{1}{68})\Gamma^{2}(\frac{9}{68})\Gamma^{2}(\frac{13}{68})\Gamma(\frac{15}{34})}{\Gamma(\frac{1}{34})\Gamma(\frac{9}{34})\Gamma(\frac{13}{34})\Gamma^{2}\left(\frac{15}{68}\right)}\right)\\

\end{align*}

1. N[EllipticK[(((3 + Sqrt[17]) (2 Sqrt[2] - Sqrt[1 + Sqrt[17]]))/
2.       8 - ((7 + Sqrt[17]) (4 - Sqrt[2 + 2 Sqrt[17]]))/
3.        32 (2 + 2 Sqrt[17])^(1/4))^2], 100]
4. N[2^(-9/2)*17^(-1/2) \[Pi]^(1/2) (Sqrt[17] - 4)^(1/8) (1 + Sqrt[17] +
5.      Sqrt[2 + 2 Sqrt[17]])^(3/2) (Product[
6.      Gamma[m/68]^KroneckerSymbol[-68, m], {m, 1, 68}])^(1/8), 100]
7. N[(6 + 2 Sqrt[17] + 5 Sqrt[4 + Sqrt[17]])^(1/4)/(
8.   2^(81/34)*17^(9/16)) ((
9.    Tan[\[Pi]/68] Tan[(9 \[Pi])/68] Tan[(13 \[Pi])/68])/
10.    Tan[(15 \[Pi])/68])^(
11.   1/4) ((Gamma[15/34] Gamma[1/68]^2 Gamma[9/68]^2 Gamma[13/68]^2)/(
12.    Gamma[1/34] Gamma[9/34] Gamma[13/34] Gamma[15/68]^2))^(1/2), 100]

复制代码

青青子衿

1. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
2. \frac{1}{\sqrt{17}}\int_{0}^{g}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-l^{2}t^{2}\right)}}dt
3. \frac{1}{\sqrt{17}}\int_{0}^{\frac{g}{\sqrt{1-g^{2}}}}\frac{1}{\sqrt{\left(1+t^{2}\right)\left(1+(1-l^{2})t^{2}\right)}}dt
4. \frac{1}{\sqrt{17}}\int_{0}^{h}\frac{1}{\sqrt{\left(1+t^{2}\right)\left(1+k^{2}t^{2}\right)}}dt
5. k=\frac{(3+\sqrt{17})(2\sqrt{2}-\sqrt{1+\sqrt{17}})}{8}-\frac{(7+\sqrt{17})(4-\sqrt{2+2\sqrt{17}})}{32}(2+2\sqrt{17})^{1/4}
6. l=\frac{(3+\sqrt{17})(2\sqrt{2}-\sqrt{1+\sqrt{17}})}{8}+\frac{(7+\sqrt{17})(4-\sqrt{2+2\sqrt{17}})}{32}(2+2\sqrt{17})^{1/4}
7. g=\frac{\sqrt{17}x\left(1+\varrho_{1}x^{2}+\varrho_{2}x^{4}+\varrho_{3}x^{6}+\varrho_{4}x^{8}+\varrho_{5}x^{10}+\varrho_{6}x^{12}+\varrho_{7}x^{14}+\varrho_{8}x^{16}\right)}{1+\rho_{1}x^{2}+\rho_{2}x^{4}+\rho_{3}x^{6}+\rho_{4}x^{8}+\rho_{5}x^{10}+\rho_{6}x^{12}+\rho_{7}x^{14}+\rho_{8}x^{16}}
8. h=\frac{\sqrt{17}x\left(1+\varrho_{1}x^{2}+\varrho_{2}x^{4}+\varrho_{3}x^{6}+\varrho_{4}x^{8}+\varrho_{5}x^{10}+\varrho_{6}x^{12}+\varrho_{7}x^{14}+\varrho_{8}x^{16}\right)}{\sqrt{1-x^{2}}\left(1+\chi_{1}x^{2}+\chi_{2}x^{4}+\chi_{3}x^{6}+\chi_{4}x^{8}+\chi_{5}x^{10}+\chi_{6}x^{12}+\chi_{7}x^{14}+\chi_{8}x^{16}\right)}
10. \varrho_{1}=-4+\frac{29\sqrt{1+\sqrt{17}}+3\sqrt{17+17\sqrt{17}}-22\sqrt{2}-2\sqrt{34}}{16}(2+2\sqrt{17})^{1/4}
11. \varrho_{2}=\frac{2361+563\sqrt{17}-720\sqrt{2+2\sqrt{17}}-174\sqrt{34+34\sqrt{17}}}{4}-\frac{203\sqrt{1+\sqrt{17}}+21\sqrt{17+17\sqrt{17}}-154\sqrt{2}-14\sqrt{34}}{32}(2+2\sqrt{17})^{1/4}
12. \varrho_{3}=-\frac{7027+1689\sqrt{17}-2160\sqrt{2+2\sqrt{17}}-522\sqrt{34+34\sqrt{17}}}{4}+\frac{20174\sqrt{2}+4910\sqrt{34}-12489\sqrt{1+\sqrt{17}}-3051\sqrt{17+17\sqrt{17}}}{16}(2+2\sqrt{17})^{1/4}
13. \varrho_{4}=\frac{1905380+462004\sqrt{17}-594871\sqrt{2+2\sqrt{17}}-144261\sqrt{34+34\sqrt{17}}}{32}-\frac{202510\sqrt{2}+49170\sqrt{34}-125905\sqrt{1+\sqrt{17}}-30615\sqrt{17+17\sqrt{17}}}{64}(2+2\sqrt{17})^{1/4}
14. \varrho_{5}=-\frac{1858608+450744\sqrt{17}-580471\sqrt{2+2\sqrt{17}}-140781\sqrt{34+34\sqrt{17}}}{16}+\frac{3834166\sqrt{2}+929954\sqrt{34}-2395093\sqrt{1+\sqrt{17}}-580939\sqrt{17+17\sqrt{17}}}{64}(2+2\sqrt{17})^{1/4}
15. \varrho_{6}=\frac{12053518+2923378\sqrt{17}-3765391\sqrt{2+2\sqrt{17}}-913241\sqrt{34+34\sqrt{17}}}{32}-\frac{11097170\sqrt{2}+2691494\sqrt{34}-6933063\sqrt{1+\sqrt{17}}-1681545\sqrt{17+17\sqrt{17}}}{128}(2+2\sqrt{17})^{1/4}
16. \varrho_{7}=-\frac{5102129+1237443\sqrt{17}-1593900\sqrt{2+2\sqrt{17}}-386578\sqrt{34+34\sqrt{17}}}{16}+\frac{7399054\sqrt{2}+1794542\sqrt{34}-4622921\sqrt{1+\sqrt{17}}-1121227\sqrt{17+17\sqrt{17}}}{64}(2+2\sqrt{17})^{1/4}
17. \varrho_{8}=\frac{267388580+64851116\sqrt{17}-83532951\sqrt{2+2\sqrt{17}}-20259733\sqrt{34+34\sqrt{17}}}{2176}-\frac{11126082\sqrt{2}+2698478\sqrt{34}-6951647\sqrt{1+\sqrt{17}}-1686025\sqrt{17+17\sqrt{17}}}{256}(2+2\sqrt{17})^{1/4}
18. \rho_{1}=\frac{170\sqrt{2}+46\sqrt{34}-85\sqrt{1+\sqrt{17}}-27\sqrt{17+17\sqrt{17}}}{16}(2+2\sqrt{17})^{1/4}
19. \rho_{2}=\frac{1326\sqrt{2+2\sqrt{17}}+324\sqrt{34+34\sqrt{17}}-4233-1027\sqrt{17}}{4}-\frac{3094\sqrt{2}+770\sqrt{34}-1853\sqrt{1+\sqrt{17}}-483\sqrt{17+17\sqrt{17}}}{32}(2+2\sqrt{17})^{1/4}
20. \rho_{3}=-\frac{46512+11248\sqrt{17}-14535\sqrt{2+2\sqrt{17}}-3489\sqrt{34+34\sqrt{17}}}{16}+\frac{18955\sqrt{1+\sqrt{17}}+4590\sqrt{17+17\sqrt{17}}-30294\sqrt{2}-7349\sqrt{34}}{4}(2+2\sqrt{17})^{1/4}
21. \rho_{4}=\frac{6328420+1534772\sqrt{17}-1977151\sqrt{2+2\sqrt{17}}-479389\sqrt{34+34\sqrt{17}}}{32}-\frac{2355401\sqrt{1+\sqrt{17}}+571071\sqrt{17+17\sqrt{17}}-3768526\sqrt{2}-914162\sqrt{34}}{64}(2+2\sqrt{17})^{1/4}
22. \rho_{5}=-\frac{838219\sqrt{2+2\sqrt{17}}+203381\sqrt{34+34\sqrt{17}}-2683484-650868\sqrt{17}}{16}+\frac{14764942\sqrt{2}+3580778\sqrt{34}-9224387\sqrt{1+\sqrt{17}}-2237469\sqrt{17+17\sqrt{17}}}{64}(2+2\sqrt{17})^{1/4}
23. \rho_{6}=\frac{26714871\sqrt{2+2\sqrt{17}}+6479425\sqrt{34+34\sqrt{17}}-85514182-20740234\sqrt{17}}{32}-\frac{146894450\sqrt{2}+35626758\sqrt{34}-91779991\sqrt{1+\sqrt{17}}-22260217\sqrt{17+17\sqrt{17}}}{128}(2+2\sqrt{17})^{1/4}
24. \rho_{7}=-\frac{18443351\sqrt{2+2\sqrt{17}}+4473257\sqrt{34+34\sqrt{17}}-59037192-14318600\sqrt{17}}{64}+\frac{3654915\sqrt{2}+886428\sqrt{34}-2283576\sqrt{1+\sqrt{17}}-553861\sqrt{17+17\sqrt{17}}}{16}(2+2\sqrt{17})^{1/4}
25. \rho_{8}=\frac{242559332+58829292\sqrt{17}-75776871\sqrt{2+2\sqrt{17}}-18378565\sqrt{34+34\sqrt{17}}}{128}-\frac{130592623\sqrt{1+\sqrt{17}}+31673337\sqrt{17+17\sqrt{17}}-209011362\sqrt{2}-50692750\sqrt{34}}{256}(2+2\sqrt{17})^{1/4}
26. \chi_{1}=-8+\frac{170\sqrt{2}+46\sqrt{34}-85\sqrt{1+\sqrt{17}}-27\sqrt{17+17\sqrt{17}}}{16}(2+2\sqrt{17})^{1/4}
27. \chi_{2}=\frac{1326\sqrt{2+2\sqrt{17}}+324\sqrt{34+34\sqrt{17}}-4121-1027\sqrt{17}}{4}-\frac{663\sqrt{1+\sqrt{17}}+105\sqrt{17\sqrt{17}+17}-714\sqrt{2}-126\sqrt{34}}{32}(2+2\sqrt{17})^{1/4}
28. \chi_{3}=-\frac{46359\sqrt{2+2\sqrt{17}}+11265\sqrt{34+34\sqrt{17}}-147208-35896\sqrt{17}}{16}+\frac{39797\sqrt{1+\sqrt{17}}+9621\sqrt{17+17\sqrt{17}}-63444\sqrt{2}-15370\sqrt{34}}{8}(2+2\sqrt{17})^{1/4}
29. \chi_{4}=\frac{5357580+1299052\sqrt{17}-1672681\sqrt{2+2\sqrt{17}}-405619\sqrt{34+34\sqrt{17}}}{32}-\frac{1275986\sqrt{2}+309582\sqrt{34}-795311\sqrt{1+\sqrt{17}}-193161\sqrt{17+17\sqrt{17}}}{64}(2+2\sqrt{17})^{1/4}
30. \chi_{5}=-\frac{14537460+3525772\sqrt{17}-4541091\sqrt{2+2\sqrt{17}}-1101349\sqrt{34+34\sqrt{17}}}{16}+\frac{24892046\sqrt{2}+6037226\sqrt{34}-15550971\sqrt{1+\sqrt{17}}-3771813\sqrt{17+17\sqrt{17}}}{64}(2+2\sqrt{17})^{1/4}
31. \chi_{6}=\frac{-7974594\sqrt{17}+10272471\sqrt{2+2\sqrt{17}}+2491465\sqrt{34+34\sqrt{17}}-32880062}{32}-\frac{14331357\sqrt{1+\sqrt{17}}+3475795\sqrt{17+17\sqrt{17}}-22933646\sqrt{2}-5562298\sqrt{34}}{128}(2+2\sqrt{17})^{1/4}
32. \chi_{7}=-\frac{62958293\sqrt{2+2\sqrt{17}}+15269643\sqrt{34+34\sqrt{17}}-201526464-48877296\sqrt{17}}{64}+\frac{23543589\sqrt{1+\sqrt{17}}+5710161\sqrt{17+17\sqrt{17}}-37680874\sqrt{2}-9138968\sqrt{34}}{32}(2+2\sqrt{17})^{1/4}
33. \chi_{8}=\frac{64851116+15728740\sqrt{17}-20259733\sqrt{2+2\sqrt{17}}-4913703\sqrt{34+34\sqrt{17}}}{128}-\frac{45874126\sqrt{2}+11126082\sqrt{34}-28662425\sqrt{\sqrt{17}+1}-6951647\sqrt{17\sqrt{17}+17}}{256}(2+2\sqrt{17})^{1/4}

复制代码

1. 1/((1 - x^2) (1 - k^2 x^2)) - (
2.     1/17 g'[x]^2)/((1 - g[x]^2) (1 - l^2 g[x]^2)) /. {
3.     k -> ( (3 + Sqrt[17]) (2 Sqrt[2] - Sqrt[1 + Sqrt[17]]))/
4.       8 - ((7 + Sqrt[17]) (4 - Sqrt[2 + 2 Sqrt[17]]))/
5.        32 (2 + 2 Sqrt[17])^(1/4),
6.     l -> ( (3 + Sqrt[17]) (2 Sqrt[2] - Sqrt[1 + Sqrt[17]]))/
7.       8 + ((7 + Sqrt[17]) (4 - Sqrt[2 + 2 Sqrt[17]]))/
8.        32 (2 + 2 Sqrt[17])^(1/4),
9.     g -> ((
10.        Sqrt[17] # (1 + Subscript[\[CurlyRho], 1] #^2 +
11.           Subscript[\[CurlyRho], 2] #^4 +
12.           Subscript[\[CurlyRho], 3] #^6 +
13.           Subscript[\[CurlyRho], 4] #^8 +
14.           Subscript[\[CurlyRho], 5] #^10 +
15.           Subscript[\[CurlyRho], 6] #^12 +
16.           Subscript[\[CurlyRho], 7] #^14 +
17.           Subscript[\[CurlyRho], 8] #^16))/(
18.        1 + Subscript[\[Rho], 1] #^2 + Subscript[\[Rho], 2] #^4 +
19.         Subscript[\[Rho], 3] #^6 + Subscript[\[Rho], 4] #^8 +
20.         Subscript[\[Rho], 5] #^10 + Subscript[\[Rho], 6] #^12 +
21.         Subscript[\[Rho], 7] #^14 + Subscript[\[Rho], 8] #^16) &)} /. {
22.    Subscript[\[CurlyRho],
23.     1] -> -4 + (
24.       29 Sqrt[1 + Sqrt[17]] + 3 Sqrt[17 + 17 Sqrt[17]] - 22 Sqrt[2] -
25.        2 Sqrt[34])/16 (2 + 2 Sqrt[17])^(1/4),
26.    Subscript[\[CurlyRho],
27.     2] -> (2361 + 563 Sqrt[17] - 720 Sqrt[2 + 2 Sqrt[17]] -
28.       174 Sqrt[34 + 34 Sqrt[17]])/4
29.      - (203 Sqrt[1 + Sqrt[17]] + 21 Sqrt[17 + 17 Sqrt[17]] -
30.        154 Sqrt[2] - 14 Sqrt[34])/32 (2 + 2 Sqrt[17])^(1/4),
31.    Subscript[\[CurlyRho],
32.     3] -> -((
33.       7027 + 1689 Sqrt[17] - 2160 Sqrt[2 + 2 Sqrt[17]] -
34.        522 Sqrt[34 + 34 Sqrt[17]])/4)
35.      + (20174 Sqrt[2] + 4910 Sqrt[34] - 12489 Sqrt[1 + Sqrt[17]] -
36.        3051 Sqrt[17 + 17 Sqrt[17]])/16 (2 + 2 Sqrt[17])^(1/4),
37.    Subscript[\[CurlyRho],
38.     4] -> (1905380 + 462004 Sqrt[17] - 594871 Sqrt[2 + 2 Sqrt[17]] -
39.       144261 Sqrt[34 + 34 Sqrt[17]])/32
40.       - (
41.       202510 Sqrt[2] + 49170 Sqrt[34] - 125905 Sqrt[1 + Sqrt[17]] -
42.        30615 Sqrt[17 + 17 Sqrt[17]])/64 (2 + 2 Sqrt[17])^(1/4),
43.    Subscript[\[CurlyRho],
44.     5] -> -((
45.       1858608 + 450744 Sqrt[17] - 580471 Sqrt[2 + 2 Sqrt[17]] -
46.        140781  Sqrt[34 + 34 Sqrt[17]])/16) + (
47.       3834166 Sqrt[2] + 929954 Sqrt[34] -
48.        2395093 Sqrt[1 + Sqrt[17]] - 580939 Sqrt[17 + 17 Sqrt[17]])/
49.       64 (2 + 2 Sqrt[17])^(1/4),
50.    Subscript[\[CurlyRho],
51.     6] -> (12053518 + 2923378 Sqrt[17] -
52.       3765391 Sqrt[2 + 2 Sqrt[17]] - 913241 Sqrt[34 + 34 Sqrt[17]])/32
53.      - (11097170 Sqrt[2] + 2691494 Sqrt[34] -
54.        6933063 Sqrt[1 + Sqrt[17]] - 1681545 Sqrt[17 + 17 Sqrt[17]])/
55.       128 (2 + 2 Sqrt[17])^(1/4),
56.    Subscript[\[CurlyRho],
57.     7] -> -((
58.       5102129 + 1237443 Sqrt[17] - 1593900 Sqrt[2 + 2 Sqrt[17]] -
59.        386578 Sqrt[34 + 34 Sqrt[17]])/16)
60.      + (7399054 Sqrt[2] + 1794542 Sqrt[34] -
61.        4622921 Sqrt[1 + Sqrt[17]] - 1121227 Sqrt[17 + 17 Sqrt[17]])/
62.       64 (2 + 2 Sqrt[17])^(1/4),
63.    Subscript[\[CurlyRho],
64.     8] -> (267388580 + 64851116 Sqrt[17] -
65.       83532951 Sqrt[2 + 2 Sqrt[17]] -
66.       20259733 Sqrt[34 + 34 Sqrt[17]])/2176
67.      - (11126082 Sqrt[2] + 2698478 Sqrt[34] -
68.        6951647 Sqrt[1 + Sqrt[17]] - 1686025 Sqrt[17 + 17 Sqrt[17]])/
69.       256 (2 + 2 Sqrt[17])^(1/4),
70.    Subscript[\[Rho],
71.     1] -> (170 Sqrt[2] + 46 Sqrt[34] - 85 Sqrt[1 + Sqrt[17]] -
72.       27  Sqrt[17 + 17 Sqrt[17]])/16  (2 + 2 Sqrt[17])^(1/4),
73.    Subscript[\[Rho],
74.     2] -> (1326 Sqrt[2 + 2 Sqrt[17]] + 324 Sqrt[34 + 34 Sqrt[17]] -
75.       4233 - 1027 Sqrt[17])/4
76.      - (3094 Sqrt[2] + 770 Sqrt[34] - 1853 Sqrt[1 + Sqrt[17]] -
77.        483 Sqrt[17 + 17 Sqrt[17]])/32 (2 + 2 Sqrt[17])^(1/4),
78.    Subscript[\[Rho],
79.     3] -> -((
80.       46512 + 11248 Sqrt[17] - 14535 Sqrt[2 + 2 Sqrt[17]] -
81.        3489 Sqrt[34 + 34 Sqrt[17]])/16)
82.      + (18955 Sqrt[1 + Sqrt[17]] + 4590 Sqrt[17 + 17 Sqrt[17]] -
83.        30294 Sqrt[2] - 7349 Sqrt[34])/4 (2 + 2 Sqrt[17])^(1/4),
84.    Subscript[\[Rho],
85.     4] -> (6328420 + 1534772 Sqrt[17] -
86.       1977151 Sqrt[2 + 2 Sqrt[17]] - 479389 Sqrt[34 + 34 Sqrt[17]])/32
87.      - (2355401 Sqrt[1 + Sqrt[17]] + 571071 Sqrt[17 + 17 Sqrt[17]] -
88.        3768526 Sqrt[2] - 914162 Sqrt[34])/64  (2 + 2 Sqrt[17])^(1/4),
89.    Subscript[\[Rho],
90.     5] -> -((
91.       838219 Sqrt[2 + 2 Sqrt[17]] + 203381 Sqrt[34 + 34 Sqrt[17]] -
92.        2683484 - 650868 Sqrt[17])/16)
93.      + (14764942 Sqrt[2] + 3580778 Sqrt[34] -
94.        9224387 Sqrt[1 + Sqrt[17]] - 2237469 Sqrt[17 + 17 Sqrt[17]])/
95.       64 (2 + 2 Sqrt[17])^(1/4),
96.    Subscript[\[Rho],
97.     6] -> (26714871 Sqrt[2 + 2 Sqrt[17]] +
98.       6479425  Sqrt[34 + 34 Sqrt[17]] - 85514182 - 20740234 Sqrt[17])/
99.      32
100.       - (
101.       146894450 Sqrt[2] + 35626758 Sqrt[34] -
102.        91779991 Sqrt[1 + Sqrt[17]] - 22260217 Sqrt[17 + 17 Sqrt[17]])/
103.       128 (2 + 2 Sqrt[17])^(1/4),
104.    Subscript[\[Rho],
105.     7] -> -((
106.       18443351 Sqrt[2 + 2 Sqrt[17]] +
107.        4473257  Sqrt[34 + 34 Sqrt[17]] - 59037192 -
108.        14318600 Sqrt[17])/64)
109.       + (
110.       3654915 Sqrt[2] + 886428 Sqrt[34] -
111.        2283576 Sqrt[1 + Sqrt[17]] - 553861 Sqrt[17 + 17 Sqrt[17]])/
112.       16 (2 + 2 Sqrt[17])^(1/4),
113.    Subscript[\[Rho],
114.     8] -> (242559332 + 58829292 Sqrt[17] -
115.       75776871 Sqrt[2 + 2 Sqrt[17]] -
116.       18378565 Sqrt[34 + 34 Sqrt[17]])/128
117.      - (130592623 Sqrt[1 + Sqrt[17]] +
118.        31673337 Sqrt[17 + 17 Sqrt[17]] - 209011362 Sqrt[2] -
119.        50692750 Sqrt[34])/256 (2 + 2 Sqrt[17])^(1/4)};
120. MinimalPolynomial[% /. x -> 1/2, x]

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青青子衿

\begin{align*}

w_{1}&=\tfrac{2947\sqrt{2}-1864\sqrt{5}+1316\sqrt{10}-4163}{3}\\

w_{2}&=\tfrac{2(2660\sqrt{2}+1050\sqrt{3}-1682\sqrt{5}-740\sqrt{6}+1189\sqrt{10}+468\sqrt{15}-332\sqrt{30}-3760)}{15}{\scriptsize(80-30\sqrt{6})^{1/3}}\\

w_{3}&=\tfrac{2(2660\sqrt{2}-1050\sqrt{3}-1682\sqrt{5}+740\sqrt{6}+1189\sqrt{10}-468\sqrt{15}+332\sqrt{30}-3760)}{15}{\scriptsize(80+30\sqrt{6})^{1/3}}\\

k(\tau={\scriptsize5\sqrt{-2}})&=w_{1}+w_{2}+w_{3}
\end{align*}

1. N[(-4163 + 2947 Sqrt[2] - 1864 Sqrt[5] + 1316 Sqrt[10])/3
2.   + (2 (-3760 + 2660 Sqrt[2] + 1050 Sqrt[3] - 1682 Sqrt[5] -
3.      740 Sqrt[6] + 1189 Sqrt[10] + 468 Sqrt[15] - 332 Sqrt[30]) (80 -
4.      30 Sqrt[6])^(1/3))/15
5.   + (2 (-3760 + 2660 Sqrt[2] - 1050 Sqrt[3] - 1682 Sqrt[5] +
6.      740 Sqrt[6] + 1189 Sqrt[10] - 468 Sqrt[15] + 332 Sqrt[30]) (80 +
7.      30 Sqrt[6])^(1/3))/15, 100]
8. N[Sqrt[ModularLambda[5 I Sqrt[2]]], 100]

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