雅可比椭圆函数虚二次复乘

青青子衿

\begin{align*}
\kappa_{2}&=\sqrt{\small{2}\>\!\scriptsize{ \big(1+\sqrt{2}\,\big)}}\\
\operatorname{sn}\left(\sqrt{-2}\,\operatorname{sn}^{-1}\left(x,\kappa_{2}i\right),\kappa_{2}i\right)&=\sqrt{-1}\cdot\frac{\sqrt{2}\,\,x \sqrt{1-x^2}}{1+\sqrt{\small{\sqrt{2}-1}}\kappa_2x^2}\\


\kappa_{3}&=2+\sqrt{3}\\
\operatorname{sn}\left(\sqrt{-3}\operatorname{sn}^{-1}\left(x,\kappa_{3}i\right),\kappa_{3}i\right)&=\sqrt{-1}\cdot\frac{x\cdot\big(\sqrt{3}-\kappa_3x^{2}\big)}{1+\sqrt{3}\kappa_3x^{2}}\\


\kappa_{4}&=2(1+\sqrt{2}\>\!)\sqrt[4]{2}\\
\operatorname{sn}\left(2\sqrt{-1}\operatorname{sn}^{-1}\left(x,\kappa_{4}i\right),\kappa_{4}i\right)&=\sqrt{-1}\cdot\frac{2\,x\cdot\big(1-\sqrt[4]{2}\,\kappa_4 x^2\big)\sqrt{1-x^2}}{1+2\sqrt[4]{8}\kappa_4x^2+(\sqrt{2}-1)\kappa_4^2x^4}\\


\kappa_{5}&=2+\sqrt{5}+2\sqrt{2+\small{\sqrt{5}}}\\
\operatorname{sn}\left(\sqrt{-5}\operatorname{sn}^{-1}\left(x,\kappa_{5}i\right),\kappa_{5}i\right)&=\sqrt{-1}\cdot\frac{x\cdot\big(\sqrt{5}-\sqrt{\small{10+10\sqrt{5}}}\,\kappa_5x^{2}+\,\kappa_5^{2}x^{4}\big)}{1+\sqrt{\small{10+10\sqrt{5}}}\,\kappa_5x^{2}+\sqrt{5}\,\kappa_5^{2}x^{4}}\\


\kappa_{6}&=\sqrt{\small{34+24 \sqrt{2}+20 \sqrt{3}+14 \sqrt{6}}}\\
\operatorname{sn}\left(\sqrt{-6}\operatorname{sn}^{-1}\left(x,\kappa_{6}i\right),\kappa_{6}i\right)
&={\scriptsize\sqrt{-1}}\cdot\tfrac{x\cdot\big({\small\sqrt{6}}-{\small{2 \sqrt{\scriptsize{4\sqrt{2}+6 \sqrt{3}+4 \sqrt{6}+6}
}}}\,\kappa_6x^2+({\small2+\sqrt{2}}\>\!) \kappa_6^2x^4\big)\sqrt{1-x^2} }
{1+\sqrt{\small{9+12 \sqrt{2}+18 \sqrt{3}+9 \sqrt{6}}}\,\kappa_6x^2
+({\small3 \sqrt{3}+\sqrt{6}}\>\!)\kappa_6^2x^4
+\sqrt{\small{2\sqrt{3}+\sqrt{6}-3-2 \sqrt{2}}\,}\kappa_6^3 x^6}
\\


\end{align*}



\begin{align*}
\operatorname{sn}\left(-2\operatorname{sn}^{-1}\left(x,\kappa_{2}i\right),\kappa_{2}i\right)&=-\,\frac{2 x }{1+\kappa _2^2 x^4}\sqrt{\big(1-x^2\big)\big(1+\kappa _2^2 x^2\big)}\\
\operatorname{sn}\left(-3\operatorname{sn}^{-1}\left(x,\kappa_{3}i\right),\kappa_{3}i\right)&=-\,\frac{x\cdot\left(3-4\left(1-\kappa_{3}^{2}\right)x^{2}-6\kappa_{3}^{2}x^{4}-\kappa_{3}^{4}x^{8}\right)}{1+6\kappa_{3}^{2}x^{4}-4\kappa_{3}^{2}\left(1-\kappa_{3}^{2}\right)x^{6}-3\kappa_{3}^{4}x^{8}}\\
\operatorname{sn}\left(-4\operatorname{sn}^{-1}\left(x,\kappa_{4}i\right),\kappa_{4}i\right)&=-\,\frac{4x\cdot\,\!P_{\operatorname{sn},4}(x)}{Q_{\operatorname{sn},4}(x)}\sqrt{\big(1-x^2\big)\big(1+\kappa _4^2 x^2\big)}\\
\operatorname{sn}\left(-5\operatorname{sn}^{-1}\left(x,\kappa_{5}i\right),\kappa_{5}i\right)&=-\,\frac{x\cdot\,\!P_{\operatorname{sn},5}(x)}{Q_{\operatorname{sn},5}(x)}\\
\end{align*}

\begin{align*}

P_{\operatorname{sn},4}(x)&=\begin{vmatrix}

x^2 & 0 & 0 & 0 & 0 & 1 \\
\small{-1} & x^2 & 0 & 0 & 0 & -2 \left(1-\kappa_4 ^2\right) \\
0 & \small{-1} & x^4 & 0 & 0 & -5 \kappa_4 ^2 \\
0 & 0 & \small{-1} & x^2 & 0 & -5 \kappa_4 ^4 \\
0 & 0 & 0 & \small{-1} & x^2 & 2 \kappa_4 ^4 \left(1-\kappa_4 ^2\right) \\
0 & 0 & 0 & 0 & \small{-1} & \kappa_4 ^6 \\

\end{vmatrix}\\
\\
Q_{\operatorname{sn},4}(x)&=\begin{vmatrix}

x^4 & 0 & 0 & 0 & 0 & 0 & 1 \\
\small{-1} & x^2 & 0 & 0 & 0 & 0 & 20 \kappa_4 ^2 \\
0 & \small{-1} & x^2 & 0 & 0 & 0 & -32 \kappa_4 ^2 \left(1-\kappa_4 ^2\right) \\
0 & 0 & \small{-1} & x^2 & 0 & 0 & 2 \kappa_4 ^2 \left(8 -29 \kappa_4 ^2+8\kappa_4 ^4\right) \\
0 & 0 & 0 & \small{-1} & x^2 & 0 & 32 \kappa_4 ^4 \left(1-\kappa_4 ^2\right) \\
0 & 0 & 0 & 0 & \small{-1} & x^4 & 20 \kappa_4 ^6 \\
0 & 0 & 0 & 0 & 0 & \small{-1} & 1 \\

\end{vmatrix}\\
\\
P_{\operatorname{sn},5}(x)&=\begin{vmatrix}

x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \\
\small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -20 \left(1-\kappa_5^2\right) \\
0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \left(8 -47 \kappa_5^2+8\kappa_5^4\right) \\
0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 80 \kappa_5^2 \left(1-\kappa_5^2\right) \\
0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & -105 \kappa_5^4 \\
0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 360 \kappa_5^4 \left(1-\kappa_5^2\right) \\
0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & -60 \kappa_5^4 \left(4 -13 \kappa_5^2+4\kappa_5^4\right) \\
0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 16 \kappa_5^4 \left(1-\kappa_5^2\right) \left(4 -31 \kappa_5^2+4\kappa_5^4\right) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 5 \kappa_5^6 \left(32-89 \kappa_5^2+32\kappa_5^4\right) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 140 \kappa_5^8 \left(1-\kappa_5^2\right) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^4 & 50 \kappa_5^{10} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & \kappa_5^{12} \\

\end{vmatrix}\\
\\
Q_{\operatorname{sn},5}(x)&=\begin{vmatrix}

x^4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 50 \kappa_5^2 \\
0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -140 \kappa_5^2 \left(1-\kappa_5^2\right) \\
0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \kappa_5^2 \left(32 -89 \kappa_5^2+32\kappa_5^4\right) \\
0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & -16 \kappa_5^2 \left(1-\kappa_5^2\right) \left(4 -31 \kappa_5^2+4\kappa_5^4\right) \\
0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & 0 & -60 \kappa_5^4 \left(4 -13 \kappa_5^2+4\kappa_5^4\right) \\
0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & 0 & -360 \kappa_5^6 \left(1-\kappa_5^2\right) \\
0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & 0 & -105 \kappa_5^8 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 0 & -80 \kappa_5^8 \left(1-\kappa_5^2\right) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 0 & 2 \kappa_5^8 \left(8 -47 \kappa_5^2+8\kappa_5^4\right) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & x^2 & 20 \kappa_5^{10} \left(1-\kappa_5^2\right) \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \small{-1} & 5 \kappa_5^{12} \\

\end{vmatrix}
\end{align*}

2. JacobiSN[
3.   Sqrt[-2] InverseJacobiSN[x, -2 (1 + Sqrt[2])], -2 (1 + Sqrt[2])] /.
4. x -> 0.123
5. (Sqrt[-2] x Sqrt[1 - x^2])/(1 + Sqrt[2] x^2) /. x -> 0.123
6. JacobiSN[Sqrt[-3]
7.    InverseJacobiSN[x, -(2 + Sqrt[3])^2], -(2 + Sqrt[3])^2]/.
8. x -> 0.123
9. (I*x (Sqrt[3] - (2 + Sqrt[3]) x^2))/(
10. 1 + Sqrt[3] (2 + Sqrt[3]) x^2) /.
11. x -> 0.123

复制代码

青青子衿

\begin{align*}
\operatorname{sn}\!\>(u,k)&=\tanh \left(\frac{\pi\,\!u}{2 K(k')}\right)\prod\limits_{n=1}^{+\infty}\frac{ \tanh \left(\frac{\pi\left(2nK(k)-u\right)}{2K(k')}\right) \tanh \left(\frac{\pi \left(2nK(k) +u\right)}{2K(k')}
\right)}
{\tanh^2\left(\frac{(2n-1)\pi\,\!K(k)}{2K(k')}\right)}\\

\operatorname{cn}\!\left(u,k\right)&=\frac{1}{\cosh\left(\frac{\pi u}{2K(k')}\right)}\prod _{n=1}^{+\infty\,}
\frac{\cosh^{2}\left(\frac{n\pi K(k)}{K(k')}\right)\sinh\left(\frac{\pi((2n-1)K(k)-u)}{2K(k')}\right)\sinh\left(\frac{\pi((2n-1)K(k)+u)}{2K(k')}\right)}{\sinh^{2}\left(\frac{(2n-1)\pi K(k)}{2K(k')}\right)\cosh\left(\frac{\pi(2nK(k)-u)}{2K(k')}\right)\cosh\left(\frac{\pi(2nK(k)+u)}{2K(k')}\right)}\\

\operatorname{dn}\!\left(u,k\right)&=\frac{1}{\cosh\left(\frac{\pi u}{2K(k')}\right)}\prod _{n=1}^{+\infty\,}
\frac{\cosh^{2}\left(\frac{n\pi K(k)}{K(k')}\right)\cosh\left(\frac{\pi((2n-1)K(k)-u)}{2K(k')}\right)\cosh\left(\frac{\pi((2n-1)K(k)+u)}{2K(k')}\right)}{\cosh^{2}\left(\frac{(2n-1)\pi K(k)}{2K(k')}\right)\cosh\left(\frac{\pi(2nK(k)-u)}{2K(k')}\right)\cosh\left(\frac{\pi(2nK(k)+u)}{2K(k')}\right)}
\end{align*}

\begin{align*}
-\frac{\tan (2 x)-i}{\tan (2 x)+i}&=\left(-\frac{\tan (x)-i}{\tan (x)+i}\right)^2\\
+\frac{\tan (3 x)-i}{\tan (3 x)+i}&=\left(+\frac{\tan (x)-i}{\tan (x)+i}\right)^3\\
-\frac{\tan (4 x)-i}{\tan (4 x)+i}&=\left(-\frac{\tan (x)-i}{\tan (x)+i}\right)^4\\
+\frac{\tan (5 x)-i}{\tan (5 x)+i}&=\left(+\frac{\tan (x)-i}{\tan (x)+i}\right)^5\\

\end{align*}


\begin{align*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{-1}^{\frac{2(1+x)^{2}-6\sqrt{1+x^{3}}}{(2-x)^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=-\int_{0}^{\frac{-2\big(\sqrt{1+x^{3}}-1\big)}{x^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{\frac{2\big(\sqrt{1+x^{3}}+1\big)}{x^{2}}}^{+\infty}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\
\\
\sqrt{3}i\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{0}^{\frac{18 x^4+2 \sqrt{3}\>\!i\>\!x\big(8-x^3\big)\sqrt{1+x^3}}{\big(4+x^3\big)^2}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\
\sqrt{3}i\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{-1}^{\frac{-(4+6 x^2+x^3)}{(x+1) (2-x)^2}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\
\sqrt{3}i\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=-\int_{0}^{\frac{18 x^4-2 \sqrt{3}\>\!i\>\!x (8-x^3) \sqrt{1+x^3}}{\big(4+x^3\big)^2}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\
\sqrt{3}i\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{-\infty}^{\frac{-(4+x^3)}{3 x^2}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\
\\
2\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{0}^{\frac{32\big(1+x^{3}\big)^{2}-4\big(8-20x^{3}-x^{6}\big)\sqrt{1+x^{3}}}{x^2\big(8-x^{3}\big)^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

2\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{-1}^{\frac{2\big(16-64x+64x^{2}+32x^{3}-56x^{4}-16x^{5}+16x^{6}+8x^{7}+x^{8}\big)-2\big(48-120x^{3}-6x^{6}\big)\sqrt{1+x^{3}}}{\big(8+8x+8x^{3}-x^{4}\big)^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

2\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=-\int_{0}^{\frac{-x\big(8-x^{3}\big)}{4\big(1+x^{3}\big)}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

2\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{\frac{32\big(1+x^{3}\big)^{2}+4\big(8-20x^{3}-x^{6}\big)\sqrt{1+x^{3}}}{x^2\big(8-x^{3}\big)^{2}}}^{+\infty}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\
\\
3\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{0}^{\frac{162x^{4}\big(4+x^{3}\big)^{4}+6x\big(4+x^{3}\big)\big(8-x^{3}\big)\big(64+48x^{3}+228x^{6}+x^{9}\big)\sqrt{1+x^{3}}}{\big(64+48x^{3}-96x^{6}+x^{9}\big)^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

3\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{-1}^{\frac{-(64-288x^{2}+48x^{3}-144x^{5}-96x^{6}-18x^{8}+x^{9})}{(x+1)(2-x)^{2}\big(x^{3}+6x^{2}+4\big)^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

3\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=-\int_{0}^{\frac{162x^{4}\big(4+x^{3}\big)^{4}-6x\big(4+x^{3}\big)\big(8-x^{3}\big)\big(64+48x^{3}+228x^{6}+x^{9}\big)\sqrt{1+x^{3}}}{\big(64+48x^{3}-96x^{6}+x^{9}\big)^{2}}}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}\\

3\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}&=\int_{\frac{64+48x^{3}-96x^{6}+x^{9}}{9x^{2}\big(4+x^{3}\big)^{2}}}^{+\infty}\frac{\mathrm{d}t}{\sqrt{1+t^{3}}}
\end{align*}

\begin{align*}
\int_{0}^{\frac{1+i}{\sqrt{2}}x}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}+\int_{0}^{\frac{1-i}{\sqrt{2}}x}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}
=\sqrt{2}\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{4}}}
\end{align*}

青青子衿

青青子衿 发表于 2023-6-10 22:51
\begin{align*}
\operatorname{sn}\!\>(u,k)&=\tanh \left(\frac{\pi\,\!u}{2 K(k')}\right)\prod\limits_{n=1}^{+\infty}\frac{ \tanh \left(\frac{\pi\left(2nK(k)-u\right)}{2K(k')}\right) \tanh \left(\frac{\pi \left(2nK(k) +u\right)}{2K(k')}
\right)}
{\tanh^2\left(\frac{(2n-1)\pi\,\!K(k)}{2K(k')}\right)}
\end{align*}

\begin{align*}
\operatorname{slh}^{-1}(x)
&=\frac{1}{2} \operatorname{sn}^{-1}\left(\frac{2 x}{1+x^2},\frac{1}{\sqrt{2}}\right)\\
&=\sqrt{2} \operatorname{sl}^{-1}\left(\frac{ x}{\sqrt{1+\sqrt{1+x^4}}}\right)\\
\int_{0}^{x}\frac{\mathrm{dt}}{\sqrt{1+t^{4}}}&=\dfrac{1}{2}\int_{0}^{\frac{2x}{1+x^2}}\frac{\mathrm{dt}}{\sqrt{(1-t^2)(1-\frac{1}{2}t^2)}}\\
&=\sqrt{2}\int_{0}^{\frac{ x}{\sqrt{1+\sqrt{1+x^4}}}}\frac{\mathrm{dt}}{\sqrt{1-t^4}}\\

\operatorname{slh}(x)&=\dfrac{\operatorname{sn}(x,\frac{1}{\sqrt{2}})}{\operatorname{cd}(x,\frac{1}{\sqrt{2}})}=\dfrac{\left(1+\operatorname{cl}^2\big(\frac{x}{\sqrt{2}}\big)\right)\operatorname{sl}\big(\frac{x}{\sqrt{2}}\big)}{\sqrt{2}\operatorname{cl}\big(\frac{x}{\sqrt{2}}\big)}\\
\operatorname{sl}(x)&=\operatorname{sc}(x,\tfrac{1}{\sqrt{2}})=\tfrac{1}{\sqrt{2}}\operatorname{sd}(\sqrt{2}\,x,\tfrac{1}{\sqrt{2}})\\
\operatorname{cl}(x)&=\operatorname{dn}(x,\tfrac{1}{\sqrt{2}})=\operatorname{cn}(\sqrt{2}\,x,\tfrac{1}{\sqrt{2}})\\


\operatorname{slh}((1+i)\operatorname{slh}^{-1}(x))&=\frac{(1+i)x}{\sqrt{1+x^4}}\\
\operatorname{slh}((1-i)\operatorname{slh}^{-1}(x))&=\frac{(1-i)x}{\sqrt{1+x^4}}\\

\operatorname{slh}(2\operatorname{slh}^{-1}(x))
&=\frac{(1+i)x}{\sqrt{1+x^4}}\circ\frac{(1-i)x}{\sqrt{1+x^4}}\\
&=\frac{(1-i)x}{\sqrt{1+x^4}}\circ\frac{(1+i)x}{\sqrt{1+x^4}}\\
&=\frac{2 x \sqrt{1+x^4}}{1-x^4}\\

\operatorname{slh}
((1+2i)\operatorname{slh}^{-1}(x))&=\frac{x \left(1+2i+x^4\right)}{1+(1+2 i) x^4}\\
\operatorname{slh}
((1-2i)\operatorname{slh}^{-1}(x))&=\frac{x \left(1-2i+x^4\right)}{1+(1-2 i) x^4}\\

\operatorname{slh}(5\operatorname{slh}^{-1}(x))&=\frac{x \left(1+2i+x^4\right)}{1+(1+2 i) x^4}\circ\frac{x \left(1-2i+x^4\right)}{1+(1-2 i) x^4}\\
&=\frac{x \left(1-2i+x^4\right)}{1+(1-2 i) x^4}\circ\frac{x \left(1+2i+x^4\right)}{1+(1+2 i) x^4}\\
&=\frac{x \left(5+2 x^4+x^8\right) \left(1+12 x^4-26 x^8-52 x^{12}+x^{16}\right)}{\left(1+2 x^4+5 x^8\right) \left(1-52 x^4-26 x^8+12 x^{12}+x^{16}\right)}\\

\operatorname{slh}((2+3i)\operatorname{slh}^{-1}(x))&=-\frac{i\,x\left(-(2+3 i)-(4-7 i) x^4-(10-11 i) x^8+i\,x^{12}\right)}
{i-(10-11 i) x^4-(4-7 i) x^8-(2+3 i) x^{12}}\\
\operatorname{slh}((2-3i)\operatorname{slh}^{-1}(x))&=\frac{i\,x\left(-(2-3 i)-(4+7 i) x^4-(10+11 i) x^8-i\,x^{12}\right)}
{-i-(10+11 i) x^4-(4+7 i) x^8-(2-3 i) x^{12}}\\
\operatorname{slh}(13\operatorname{slh}^{-1}(x))&=\operatorname{slh}((2+3i)\operatorname{slh}^{-1}(x))\circ\operatorname{slh}((2-3i)\operatorname{slh}^{-1}(x))\\
&=\operatorname{slh}((2-3i)\operatorname{slh}^{-1}(x))\circ\operatorname{slh}((2+3i)\operatorname{slh}^{-1}(x))\\


\end{align*}

1. JacobiSN[(2 + 3 I) 1/2 InverseJacobiSN[(2 x)/(1 + x^2), 1/2], 1/2]/
2. JacobiCD[(2 + 3 I) 1/2 InverseJacobiSN[(2 x)/(1 + x^2), 1/2], 1/
3.   2] /. x -> 0.4
4. (-I*x (-(2 + 3 I) - (4 - 7 I ) x^4 - (10 - 11 I ) x^8 +
5.     I*x^12))/(I - (10 - 11 I ) x^4 - (4 - 7 I ) x^8 - (2 +
6.       3 I ) x^12) /. x -> 0.4
7. (u Sqrt[1 + v^4] + v Sqrt[1 + u^4])/(
8.   1 - u^2 v^2) /. {u -> (2 x Sqrt[1 + x^4])/(1 - x^4),
9.    v -> (x (3 + 6 x^4 - x^8))/(1 - 6 x^4 - 3 x^8)*I} /. x -> 0.4

复制代码

\begin{align*}
\operatorname{slh}_3^{-1}(x)
&=\dfrac{1}{\sqrt[4]{3}}\operatorname{sn}^{-1}\left(\frac{2\sqrt[4]{3}x\left(\sqrt{1+x}+\sqrt{1-x+x^{2}}\right)}{\big(1+\sqrt{3}\,\big)x^{2}+2\big(\,1+\sqrt{1+x^{3}}\,\big)},\frac{\sqrt{2}+\sqrt{6}}{4}\right)\\
&= \operatorname{sl}_3^{-1}\left(\frac{2\big(\sqrt{1+x^{3}}-1\big)}{x^{2}}\right)\\


\end{align*}

2. \rho_{a}\left(x,r\right)=\frac{x\left(3+3\sqrt{2}-2\sqrt{3}-\sqrt{6}+\left(10+6\sqrt{2}-5\sqrt{3}-4\sqrt{6}\right)x^{2}\right)}{1+\left(12+9\sqrt{2}-7\sqrt{3}-5\sqrt{6}\right)x^{2}}
3. *****
4. \rho_{b}\left(x,r\right)=\frac{\left(2\sqrt{3}+\sqrt{6}-2-2\sqrt{2}\right)x}{1+\left(2\sqrt{3}+\sqrt{6}-3-2\sqrt{2}\right)x^{2}}
5. *****
6. \int_{0}^{\rho_{a}\left(x,r\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\left(\sqrt{3}-2\right)\left(\sqrt{3}+\sqrt{2}\right)t\right)^{2}\right)}}dt
7. *****
8. \sqrt{\frac{3}{2}}\int_{0}^{\rho_{b}\left(x,r\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\sqrt{24\sqrt{2}+20\sqrt{3}-34-14\sqrt{6}}t\right)^{2}\right)}}dt
9. *****
10. \kappa_{r}=\sqrt{34-24\sqrt{2}+20\sqrt{3}-14\sqrt{6}}
11. *****
12. \psi_{a}\left(x,r\right)=\frac{\rho_{a}\left(x,r\right)}{\sqrt{1+\kappa_{r}^{2}-\rho_{a}\left(x,r\right)^{2}}}
13. *****
14. \psi_{b}\left(x,r\right)=\frac{\rho_{b}\left(x,r\right)}{\sqrt{1+\kappa_{r}^{2}-\kappa_{r}^{2}\rho_{b}\left(x,r\right)^{2}}}
15. *****
16. \int_{0}^{\psi_{a}\left(x,r\right)}\frac{1}{\sqrt{\left(1+t^{2}\right)\left(1-\left(\kappa_{r}t\right)^{2}\right)}}dt
17. *****
18. \sqrt{\frac{3}{2}}\int_{0}^{\psi_{b}\left(x,r\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1+\left(\kappa_{r}t\right)^{2}\right)}}dt

复制代码

青青子衿

青青子衿 发表于 2023-7-3 12:57

\begin{align*}
&\qquad{\Large\int}_{0}^{1}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3})^{2}(\sqrt{3}-\sqrt{2}) ^2t^{2}\big)}}\\
&=\frac{1}{\sqrt{6}\,}{\Large\int}_{0}^{1}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\!\>)}}\>\!)^{2}t^{2}\big)}}\\
\end{align*}

\begin{align*}


&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{2}-1)^{2}t^{2}\big)}}\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\rho(x,\sqrt{2}\>\!)
}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\\
&\qquad\qquad\quad\rho(x,\sqrt{2}\>\!)=
\small{\frac{\sqrt{2}\,x}{1+\big(\!\sqrt{2}-1\big)x^{2}}}\\
\\

&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\chi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{2}-1)^{2}t^{2}\big)}}\\
\\
&\qquad\>\>\>\chi(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\,x\sqrt{(1-x^{2})(1-2(\sqrt{2}-1)x^{2})}}{1-2(\sqrt{2}-1)x^{2}}\\
\\

&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(1+\sqrt{2}\>\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\varphi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(1+\sqrt{2}\>\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\\
&\qquad\qquad\quad\>\>\>\varphi(x,\sqrt{2}\>\!)=
\tfrac{\sqrt{2}\,x\sqrt{1-x^{2}}}{1+\sqrt{2}\>\!x^{2}}\\
\\

&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\psi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\\
&\qquad\qquad\quad\>\>\>\psi(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\,x\sqrt{1+x^{2}}}{1-\sqrt{2}\>\!x^{2}}
\\

\end{align*}


\begin{align*}
&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-\big(3-2\sqrt{2}\>\!\big)^{2}t^{2}\big)}}\\
&=\frac{1}{2}{\Large\int}_{0}^{\rho(x,2)
}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-\big(\small{2 \sqrt[4]{2}\>(\sqrt{2}-1)}\big)^2t^{2}\big)}}\\
\\
&\qquad\quad\rho(x,2)=
\tfrac{2x\cdot\big(1+(3-2 \sqrt{2}\>\!) x^2\big)}{1+2(4 \sqrt{2}-5) x^2+(17-12\sqrt{2}\>\!) x^4}\\
\\

&\qquad{\Large\int}_{0}^{x}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-\big(\small{2 \sqrt[4]{2}\>(\sqrt{2}-1)}\big)^2t^{2}\big)}}\\
&=\frac{1}{2}{\Large\int}_{0}^{
\chi(x,2)
}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-\big(3-2\sqrt{2}\>\!\big)^{2}t^{2}\big)}}\\
\\
&\quad\>\chi(x,2)=
\tfrac{2x\cdot\big(1-2(2-\sqrt{2}\>\!)x^{2}\big)\sqrt{\small{
\big(1-x^{2}\big)\big(1-4(3\sqrt{2}-4)x^{2}\big)}}
}{
1+2(9-7\sqrt{2}\>\!)x^{2}+8(10-7\sqrt{2}\>\!)x^{4}
}\\
\\

&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+\big(\small{2 \sqrt[4]{2}\>(1+\sqrt{2}\>\!)}\big)^2t^{2}\big)}}\\
&=\frac{1}{2}{\Large\int}_{0}^{\varphi(x,2)
}
\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-\big(\small{2 \sqrt[4]{2}\>(1+\sqrt{2}\>\!)}\big)^2t^{2}\big)}}\\
\\
&\qquad\qquad\varphi(x,2)=
\tfrac{2x\cdot\big(1-2(2+\sqrt{2}\>\!) x^2\big)\sqrt{1-x^2}}{1+8 (1+\sqrt{2}\>\!) x^2+4(2+\sqrt{2}\>\!) x^4}\\
\\

&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-\big(\small{2 \sqrt[4]{2}\>(1+\sqrt{2}\>\!)}\big)^2t^{2}\big)}}\\
&=\frac{1}{2}{\Large\int}_{0}^{\psi(x,2)
}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+\big(\small{2 \sqrt[4]{2}\>(1+\sqrt{2}\>\!)}\big)^2t^{2}\big)}}\\
\\
&\qquad\qquad\>\psi(x,2)=
\tfrac{2x\cdot\big(1+2 (2+\sqrt{2}\>\!)x^2\big)\sqrt{1+x^2}}{1-8 (1+\sqrt{2}\>\!) x^2+4(2+\sqrt{2}\>\!)x^4}\\

\end{align*}





\begin{align*}
&\qquad{\Large\int}_{0}^{x}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3})^{2}(\sqrt{3}-\sqrt{2}) ^2t^{2}\big)}}\\

&=\frac{1}{\sqrt{6}\,}{\Large\int}_{0}^{\rho(x,\sqrt{6}\>\!\>\!)}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\!\>)}}\>\!)^{2}t^{2}\big)}}\\
\\
\rho(x,\sqrt{6}\>\!\>\!)&=\tfrac{x\cdot\big(\sqrt{6}+(20\sqrt{3}+12\sqrt{6}-32-22\sqrt{2}\>\!)x^{2}+(48\sqrt{3}+35\sqrt{6}-84-60\sqrt{2}\>\!)x^{4}\big)}{1+(16\sqrt{3}+11\sqrt{6}-27-18\sqrt{2}\>\!)x^{2}+(315+222\sqrt{2}-182\sqrt{3}-128\sqrt{6}\>\!)x^{4}+(234\sqrt{3}+165\sqrt{6}-405-286\sqrt{2}\>\!)x^{6}}\\
\\
&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\!\>)}}\>\!)^{2}t^{2}\big)}}\\

&=\frac{1}{\sqrt{6}\,}{\Large\int}_{0}^{\chi(x,\sqrt{6}\>\!\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3})^{2}(\sqrt{3}-\sqrt{2}) ^2t^{2}\big)}}\\
\\
\chi(x,\sqrt{6}\>\!\>\!)&=\tfrac{x\cdot\big(\sqrt{6}-(2+\sqrt{2}\>\!)(\sqrt{3}-1)x^{2}\big)\big(1-(13+9\sqrt{2}-7\sqrt{3}-5\sqrt{6}\>\!)x^{2}\big)\sqrt{(1-x^{2})(1-(\!\sqrt{\small{2(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\>\!)}}\>\!)^2x^{2})}}
{\big(1-(\sqrt{3}-1)x^{2}\big)\big(1-2(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\>\!)x^{2}\big)\big(1-(5\sqrt{3}+4\sqrt{6}-9-6\sqrt{2}\>\!)x^{2}\big)}\\
\\
&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(17+12\sqrt{2}+10\sqrt{3}+7\sqrt{6})}}\>\!)^{2}t^{2}\big)}}\\
&=\dfrac{1}{\sqrt{6}\,}{\Large\int}_{0}^{\varphi(x,\sqrt{6}\>\!\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(17+12\sqrt{2}+10\sqrt{3}+7\sqrt{6}\>\!)}}\>\!)^{2}t^{2}\big)}}\\
\\
\varphi(x,\sqrt{6}\>\!\>\!)&=\tfrac{x\cdot\big(\sqrt{6}-(2+\sqrt{2}\>\!) (1+\sqrt{3}\>\!)x^2\big)\big(1-(13+9 \sqrt{2}+7 \sqrt{3}+5 \sqrt{6}\>\!) x^2\big)\sqrt{1-x^{2}}}{1+\big(24+18\sqrt{2}+16\sqrt{3}+11\sqrt{6}\>\!\big)x^{2}+2\big(132+93\sqrt{2}+75\sqrt{3}+53\sqrt{6}\>\!\big)x^{4}+2\big(58+41\sqrt{2}+34\sqrt{3}+24\sqrt{6}\>\!\big)x^{6}}\\
\\
&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(17+12\sqrt{2}+10\sqrt{3}+7\sqrt{6})}}\>\!)^{2}t^{2}\big)}}\\
&=\dfrac{1}{\sqrt{6}\,}{\Large\int}_{0}^{\psi(x,\sqrt{6}\>\!\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(17+12\sqrt{2}+10\sqrt{3}+7\sqrt{6}\>\!)}}\>\!)^{2}t^{2}\big)}}\\
\\
\psi(x,\sqrt{6}\>\!\>\!)&=\tfrac{x\cdot\big(\sqrt{6}+(2+\sqrt{2}\>\!) (1+\sqrt{3}\>\!)x^2\big)\big(1+(13+9 \sqrt{2}+7 \sqrt{3}+5 \sqrt{6}\>\!) x^2\big)\sqrt{1+x^{2}}}{1-\big(24+18\sqrt{2}+16\sqrt{3}+11\sqrt{6}\>\!\big)x^{2}+2\big(132+93\sqrt{2}+75\sqrt{3}+53\sqrt{6}\>\!\big)x^{4}-2\big(58+41\sqrt{2}+34\sqrt{3}+24\sqrt{6}\>\!\big)x^{6}}\\
\\

\end{align*}


\begin{align*}
\kappa_6&=\sqrt{\small{34+24\sqrt{2}+20\sqrt{3}+14\sqrt{6}}}\\
\\
\rho(x,\sqrt{r}\>\!\>\!)&=\frac{\sqrt{\small{1+\kappa_{r}^2}}\,\psi\big(\left.x \middle/{\sqrt{\small{1+\kappa_{r}^{2}-x^{2}}}}\right.,\sqrt{r}\>\!\>\!\big)}{\sqrt{1+\kappa_{r}^{2}\>\!\psi^2\big(\left.x \middle/{\sqrt{\small{1+\kappa_{r}^{2}-x^2}}}\right.,\sqrt{r}\>\!\>\!\big)\,}}\\
\\
\chi(x,\sqrt{r}\>\!\>\!)&=\frac{\sqrt{\small{1+\kappa_{r}^2}}\,\varphi\big(\left.x \middle/{\sqrt{\small{1+\kappa_{r}^{2}-\kappa_{r}^{2}x^{2}}}}\right.,\sqrt{r}\>\!\>\!\big)}{\sqrt{1+\varphi^2\big(\left.x \middle/{\sqrt{\small{1+\kappa_{r}^{2}-\kappa_{r}^{2}x^2}}}\right.,\sqrt{r}\>\!\>\!\big)\,}}\\
\\
\varphi(x,\sqrt{r}\>\!\>\!)&=\frac{\chi\big(\left.\sqrt{\small{1+\kappa_{r}^2}}\,x \middle/{\sqrt{\small{1+\kappa_{r}^2x^2}}}\right.,\sqrt{r}\>\!\>\!\big)}{\sqrt{1+\kappa_{r}^{2}-\chi^2\big(\left.\sqrt{\small{1+\kappa_{r}^2}}\,x \middle/{\sqrt{\small{1+\kappa_{r}^2x^2}}}\right.,\sqrt{r}\>\!\>\!\big)\,}}\\
\\
\psi(x,\sqrt{r}\>\!\>\!)&=\frac{\rho\big(\left.\sqrt{\small{1+\kappa_{r}^2}}\,x \middle/{\sqrt{\small{1+x^2}}}\right.,\sqrt{r}\>\!\>\!\big)}{\sqrt{1+\kappa_{r}^{2}-\kappa_{r}^{2}\>\!\rho^2\big(\left.\sqrt{\small{1+\kappa_{r}^2}}\,x \middle/{\sqrt{\small{1+x^2}}}\right.,\sqrt{r}\>\!\>\!\big)\,}}\\

\end{align*}

2. \rho\left(x,\Omega\right)=\frac{x\left(\sqrt{6}+\left(20\sqrt{3}+12\sqrt{6}-32-22\sqrt{2}\right)x^{2}+\left(48\sqrt{3}+35\sqrt{6}-84-60\sqrt{2}\right)x^{4}\right)}{1+\left(16\sqrt{3}+11\sqrt{6}-27-18\sqrt{2}\right)x^{2}+\left(315+222\sqrt{2}-182\sqrt{3}-128\sqrt{6}\right)x^{4}+\left(234\sqrt{3}+165\sqrt{6}-405-286\sqrt{2}\right)x^{6}}
3. 00000
4. \chi\left(x,\Omega\right)=\frac{x\left(\sqrt{6}-\left(2+\sqrt{2}\right)\left(\sqrt{3}-1\right)x^{2}\right)\left(1-\left(13+9\sqrt{2}-7\sqrt{3}-5\sqrt{6}\right)x^{2}\right)\sqrt{\left(1-x^{2}\right)\left(1-2\left(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\right)x^{2}\right)}}{\left(1-\left(\sqrt{3}-1\right)x^{2}\right)\left(1-2\left(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\right)x^{2}\right)\left(1-\left(5\sqrt{3}+4\sqrt{6}-9-6\sqrt{2}\right)x^{2}\right)}
5. 00000
6. \varphi\left(x,\Omega\right)=\frac{x\left(\sqrt{6}-\left(2+\sqrt{2}\right)\left(1+\sqrt{3}\right)x^{2}\right)\left(1-\left(13+9\sqrt{2}+7\sqrt{3}+5\sqrt{6}\right)x^{2}\right)\sqrt{1-x^{2}}}{1+\left(24+18\sqrt{2}+16\sqrt{3}+11\sqrt{6}\right)x^{2}+2\left(132+93\sqrt{2}+75\sqrt{3}+53\sqrt{6}\right)x^{4}+2\left(58+41\sqrt{2}+34\sqrt{3}+24\sqrt{6}\right)x^{6}}
7. 00000
8. \psi\left(x,\Omega\right)=\frac{x\left(\sqrt{6}+\left(2+\sqrt{2}\right)\left(1+\sqrt{3}\right)x^{2}\right)\left(1+\left(13+9\sqrt{2}+7\sqrt{3}+5\sqrt{6}\right)x^{2}\right)\sqrt{1+x^{2}}}{1-\left(24+18\sqrt{2}+16\sqrt{3}+11\sqrt{6}\right)x^{2}+2\left(132+93\sqrt{2}+75\sqrt{3}+53\sqrt{6}\right)x^{4}-2\left(58+41\sqrt{2}+34\sqrt{3}+24\sqrt{6}\right)x^{6}}
9. 00000
10. \kappa_{6}=\sqrt{34+24\sqrt{2}+20\sqrt{3}+14\sqrt{6}}
11. 00000
12. \rho\left(x,\sqrt{6}\right)-\frac{\sqrt{1+\kappa_{6}^{2}}\psi\left(\frac{x}{\sqrt{1+\kappa_{6}^{2}-x^{2}}},\sqrt{6}\right)}{\sqrt{1+\kappa_{6}^{2}\psi\left(\frac{x}{\sqrt{1+\kappa_{6}^{2}-x^{2}}},\sqrt{6}\right)^{2}}}
13. 00000
14. \chi\left(x,\sqrt{6}\right)-\frac{\sqrt{1+\kappa_{6}^{2}}\varphi\left(\frac{x}{\sqrt{1+\kappa_{6}^{2}-\kappa_{6}^{2}x^{2}}},\sqrt{6}\right)}{\sqrt{1+\varphi\left(\frac{x}{\sqrt{1+\kappa_{6}^{2}-\kappa_{6}^{2}x^{2}}},\sqrt{6}\right)^{2}}}
15. 00000
16. \varphi\left(x,\sqrt{6}\right)-\frac{\chi\left(\frac{\sqrt{1+\kappa_{6}^{2}}x}{\sqrt{1+\kappa_{6}^{2}x^{2}}},\sqrt{6}\right)}{\sqrt{1+\kappa_{6}^{2}-\chi\left(\frac{\sqrt{1+\kappa_{6}^{2}}x}{\sqrt{1+\kappa_{6}^{2}x^{2}}},\sqrt{6}\right)^{2}}}
17. 00000
18. \psi\left(x,\sqrt{6}\right)-\frac{\rho\left(\frac{\sqrt{1+\kappa_{6}^{2}}x}{\sqrt{1+x^{2}}},\sqrt{6}\right)}{\sqrt{1+\kappa_{6}^{2}-\kappa_{6}^{2}\rho\left(\frac{\sqrt{1+\kappa_{6}^{2}}x}{\sqrt{1+x^{2}}},\sqrt{6}\right)^{2}}}
19. 00000
20. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\left(2-\sqrt{3}\right)\left(\sqrt{3}-\sqrt{2}\right)t\right)^{2}\right)}}dt
21. 00000
22. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\sqrt{20\sqrt{3}+14\sqrt{6}-34-24\sqrt{2}}t\right)^{2}\right)}}dt
23. 00000
24. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1+\left(\sqrt{34+24\sqrt{2}+20\sqrt{3}+14\sqrt{6}}t\right)^{2}\right)}}dt
25. 00000
26. \int_{0}^{x}\frac{1}{\sqrt{\left(1+t^{2}\right)\left(1-\left(\sqrt{34+24\sqrt{2}+20\sqrt{3}+14\sqrt{6}}t\right)^{2}\right)}}dt
27. 00000
28. \frac{1}{\sqrt{6}}\int_{0}^{\rho\left(x,\sqrt{6}\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\sqrt{20\sqrt{3}+14\sqrt{6}-34-24\sqrt{2}}t\right)^{2}\right)}}dt
29. 00000
30. \frac{1}{\sqrt{6}}\int_{0}^{\chi\left(x,\sqrt{6}\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\left(\sqrt{3}-2\right)\left(\sqrt{3}-\sqrt{2}\right)t\right)^{2}\right)}}dt
31. 00000
32. \frac{1}{\sqrt{6}}\int_{0}^{\varphi\left(x,\sqrt{6}\right)}\frac{1}{\sqrt{\left(1+t^{2}\right)\left(1-\left(\sqrt{34+24\sqrt{2}+20\sqrt{3}+14\sqrt{6}}t\right)^{2}\right)}}dt
33. 00000
34. \frac{1}{\sqrt{6}}\int_{0}^{\psi\left(x,\sqrt{6}\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1+\left(\sqrt{34+24\sqrt{2}+20\sqrt{3}+14\sqrt{6}}t\right)^{2}\right)}}dt

复制代码

\begin{align*}
F(x)&=\frac{f(x)\cdot\left[3 \sqrt{2}+2 \sqrt{3}-3-\sqrt{6}+(13-9\sqrt{2}-7 \sqrt{3}+5\sqrt{6}\>\>\!\!)\cdot\,\!f^2(x)\right]}{1+(9-6\sqrt{2}-5\sqrt{3}+4\sqrt{6}\>\>\!\!) \cdot\,\!f^2(x)}\\
&=
\tfrac{x\cdot\big(\sqrt{6}+(20\sqrt{3}+12\sqrt{6}-32-22\sqrt{2}\>\!)x^{2}+(48\sqrt{3}+35\sqrt{6}-84-60\sqrt{2}\>\!)x^{4}\big)}{1+(16\sqrt{3}+11\sqrt{6}-27-18\sqrt{2}\>\!)x^{2}+(315+222\sqrt{2}-182\sqrt{3}-128\sqrt{6}\>\!)x^{4}+(234\sqrt{3}+165\sqrt{6}-405-286\sqrt{2}\>\!)x^{6}}\\
\\
\end{align*}

1. \frac{y\left(3\sqrt{2}+2\sqrt{3}-3-\sqrt{6}+\left(13-9\sqrt{2}-7\sqrt{3}+5\sqrt{6}\right)y^{2}\right)}{1+\left(9-6\sqrt{2}-5\sqrt{3}+4\sqrt{6}\right)y^{2}}=\frac{x\left(\sqrt{6}+\left(20\sqrt{3}+12\sqrt{6}-32-22\sqrt{2}\right)x^{2}+\left(48\sqrt{3}+35\sqrt{6}-84-60\sqrt{2}\right)x^{4}\right)}{1+\left(16\sqrt{3}+11\sqrt{6}-27-18\sqrt{2}\right)x^{2}+\left(315+222\sqrt{2}-182\sqrt{3}-128\sqrt{6}\right)x^{4}+\left(234\sqrt{3}+165\sqrt{6}-405-286\sqrt{2}\right)x^{6}}
3. with(algcurves);
4. f := y*(3*sqrt(2) + 2*sqrt(3) - 3 - sqrt(6) + (13 - 9*sqrt(2) - 7*sqrt(3) + 5*sqrt(6))*y^2)*(1 + (234*sqrt(3) + 165*sqrt(6) - 405 - 286*sqrt(2))*x^6 + (315 + 222*sqrt(2) - 182*sqrt(3) - 128*sqrt(6))*x^4 + (16*sqrt(3) + 11*sqrt(6) - 27 - 18*sqrt(2))*x^2) - (1 + (9 - 6*sqrt(2) - 5*sqrt(3) + 4*sqrt(6))*y^2)*x*(sqrt(6) + (20*sqrt(3) + 12*sqrt(6) - 32 - 22*sqrt(2))*x^2 + (48*sqrt(3) + 35*sqrt(6) - 84 - 60*sqrt(2))*x^4);
5. genus(f, x, y);

复制代码

青青子衿

青青子衿 发表于 2023-7-3 12:57

\begin{align*}

&\qquad{\Large\int}_{0}^{\frac{535+2169\sqrt{2}-774\sqrt{3}+315\sqrt{6}}{4946}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{2}+\sqrt{3}\>\>\!\!) ^2t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\frac{2(212+102\sqrt{2}-42\sqrt{3}-69\sqrt{6}\>\!)}{431}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\

&\qquad{\Large\int}_{0}^{\frac{x\cdot(3+3\sqrt{2}-2\sqrt{3}-\sqrt{6}+(10+6\sqrt{2}-5\sqrt{3}-4\sqrt{6}\>\!)x^{2})}{1+(12+9\sqrt{2}-7\sqrt{3}-5\sqrt{6}\>\!)x^{2}}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{2}+\sqrt{3}\>\>\!\!) ^2t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\frac{(2\sqrt{3}+\sqrt{6}-2-2\sqrt{2}\>\!)x}{1+(2\sqrt{3}+\sqrt{6}-3-2\sqrt{2}\!\>)x^{2}}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\

\end{align*}

\begin{align*}
&\qquad{\Large\int}_{0}^{\rho_a(x,\sqrt{\small{2/3}}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{2}+\sqrt{3}\>\>\!\!) ^2t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\rho_b(x,\sqrt{\small{2/3}}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\\
&\quad\>\>\rho_a(x,\sqrt{\small{2/3}}\>\!)=\tfrac{(\sqrt{3}-\sqrt{2}\>\!)x\cdot(\sqrt{3}+\sqrt{6}+(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2})}{1+(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)(\sqrt{3}+\sqrt{6}\>\!)x^{2}}\\
&\quad\>\>\rho_b(x,\sqrt{\small{2/3}}\>\!)=\tfrac{(2+\sqrt{2}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x}{1+(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2}}\\
\\
&\qquad{\Large\int}_{0}^{\chi_a(x,\sqrt{\small{2/3}}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\chi_b(x,\sqrt{\small{2/3}}\>\!)}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{2}+\sqrt{3}\>\>\!\!) ^2t^{2}\big)}}
\\
\\
&\quad\>\>\chi_a(x,\sqrt{\small{2/3}}\>\!)=\tfrac{(\sqrt{3}-\sqrt{2}\>\!)x\cdot(\sqrt{3}+\sqrt{6}-(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2})\sqrt{1+x^{2}}}{(1+x^{2})(1+(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)^{2}x^{2})}\\
&\quad\>\>\chi_b(x,\sqrt{\small{2/3}}\>\!)=\tfrac{(2+\sqrt{2}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x\sqrt{(1+x^{2})(1+(2-\sqrt{3}\>\!)^{2}(\sqrt{3}-\sqrt{2}\>\!)^{2}x^{2})}}{(1+x^{2})(1+(2-\sqrt{3}\>\!)^{2}(\sqrt{3}-\sqrt{2}\>\!)^{2}x^{2})}\\
\\
&\qquad{\Large\int}_{0}^{\varphi_a(x,\sqrt{\small{2/3}}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(17-12\sqrt{2}+10\sqrt{3}-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\varphi_a(x,\sqrt{\small{2/3}}\>\!)}
\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(17-12\sqrt{2}+10\sqrt{3}-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\\
&\quad\>\>\varphi_a(x,\sqrt{\small{2/3}}\>\!)=
\tfrac{(2-\sqrt{3}\>\!)x\cdot(\sqrt{3}+\sqrt{6}-(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2})\sqrt{1+(2-\sqrt{3}\>\!)^{2}(\sqrt{3}-\sqrt{2}\>\!)^{2}x^{2}}}
{(1+(2-\sqrt{3}\>\!)x^{2})(1+(2-\sqrt{3}\>\!)^{2}(\sqrt{3}-\sqrt{2}\!\>)^{2}x^{2})}\\
&\quad\>\>\varphi_b(x,\sqrt{\small{2/3}}\>\!)=\tfrac{(2+\sqrt{2}\>\!)(2-\sqrt{3}\>\!)x}{1+(2+\sqrt{2}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2}}\\
\\
&\quad{\Large\int}_{0}^{\psi_a(x,\sqrt{\small{2/3}}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(17-12\sqrt{2}+10\sqrt{3}-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\psi_b(x,\sqrt{\small{2/3}}\>\!)}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(17-12\sqrt{2}+10\sqrt{3}-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\\
&\quad\>\>\psi_b(x,\sqrt{\small{2/3}}\>\!)=
\tfrac{(2-\sqrt{3}\>\!)x\cdot(\sqrt{3}+\sqrt{6}+(2-\sqrt{3}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2})\sqrt{1-(2-\sqrt{3}\>\!)^{2}(\sqrt{3}-\sqrt{2}\>\!)^{2}x^{2}}}
{(1-(2-\sqrt{3}\>\!)x^{2})(1-(2-\sqrt{3}\>\!)^{2}(\sqrt{3}-\sqrt{2}\!\>)^{2}x^{2})}\\
&\quad\>\>\psi_b(x,\sqrt{\small{2/3}}\>\!)=\tfrac{(2+\sqrt{2}\>\!)(2-\sqrt{3}\>\!)x}{1-(2+\sqrt{2}\>\!)(\sqrt{3}-\sqrt{2}\>\!)x^{2}}\\
\\
\end{align*}


\begin{align*}
&\quad{\Large\int}_{0}^{\frac{x(3+3\sqrt{2}-2\sqrt{3}-\sqrt{6}+(7\sqrt{3}+5\sqrt{6}-13-9\sqrt{2}\!\>)x^{2})}{1+(9+6\sqrt{2}-5\sqrt{3}-4\sqrt{6}\>\!)x^{2}}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\left(3+3\sqrt{2}-2\sqrt{3}-\sqrt{6}\>\!\right){\Large\int}_{0}^{x}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(7\sqrt{6}+10\sqrt{3}-17-12\sqrt{2}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\
\end{align*}

青青子衿

青青子衿 发表于 2023-8-15 19:10
\begin{align*}

&\qquad{\Large\int}_{0}^{\frac{535+2169\sqrt{2}-774\sqrt{3}+315\sqrt{6}}{4946}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{2}+\sqrt{3}\>\>\!\!) ^2t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\frac{2(212+102\sqrt{2}-42\sqrt{3}-69\sqrt{6}\>\!)}{431}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\

&\qquad{\Large\int}_{0}^{\frac{x\cdot(3+3\sqrt{2}-2\sqrt{3}-\sqrt{6}+(10+6\sqrt{2}-5\sqrt{3}-4\sqrt{6}\>\!)x^{2})}{1+(12+9\sqrt{2}-7\sqrt{3}-5\sqrt{6}\>\!)x^{2}}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{2}+\sqrt{3}\>\>\!\!) ^2t^{2}\big)}}\\
&=\sqrt{\frac{3}{2}}{\Large\int}_{0}^{\frac{(2\sqrt{3}+\sqrt{6}-2-2\sqrt{2}\>\!)x}{1+(2\sqrt{3}+\sqrt{6}-3-2\sqrt{2}\!\>)x^{2}}}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(12\sqrt{2}+10\sqrt{3}-17-7\sqrt{6}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\

\end{align*}

\begin{align*}
j\big(i\sqrt{\small{\left.2\middle/3\right.}}\>\!\big)=12^3\big(1-\sqrt{2}\>\!\big)^2\big(5-2 \sqrt{2}\>\!\big)^3
\end{align*}

\begin{align*}
\sqrt{7}\int_{x}^{+\infty}\frac{\mathrm{d}t}{\sqrt{t^{3}-35t-98}}=
{\large\int}_{\scriptsize-\frac{422576+285719x+120050x^{2}+24353x^{3}+588x^{4}-343x^{5}-14x^{6}-x^{7}}{7(91+21x-7x^{2}-x^{3})^{2}}}^{+\infty}\frac{\mathrm{d}t}{\sqrt{t^{3}-35t+98}}
\end{align*}

\begin{align*}
&\qquad\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\scriptsize{\frac{3\sqrt{2}-\sqrt{14}}{8}}})^{2}t^2)}}\\
&\qquad=\frac{1}{\sqrt{7}\,}{\Large\int}_{0}^{\rho(x,\sqrt{7}\>\!)}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\scriptsize{\frac{3\sqrt{2}+\sqrt{14}}{8}}})^{2}t^2)}}\\
\\
&\rho(x,\sqrt{7}\>\!)=\tfrac{x(64\sqrt{7}+48(7-2\sqrt{7}\>\!)x^{2}+16(8\sqrt{7}-21)x^{4}+(127-48\sqrt{7}\>\!)x^{6})}{64+64\sqrt{7}x^{2}+(84-24\sqrt{7}\>\!)x^{4}+(8\sqrt{7}-21)x^{6}}\\
\\
\end{align*}

1. y^2 - (x^3 - 30*x + 56) /. {
2.    x -> -(1/2) (x + (72 (x - 4))/(2 (x - 4))^2),
3.    y -> -(y/(2 Sqrt[-2])) (1 - (144 (x - 4))/(2 (x - 4))^3)} // Factor
4. -((18 - 4 X + X^2)/(2 (X - 4))) - (
5.    x*4 (x^3 - 30*x + 56) - 3 (x^2 - 4 x + 6) (x^2 + 4 x - 50))/(
6.    4 (x - 4) (x^2 + 4 x - 14)) /.
7.   X -> -((18 - 4 x + x^2)/(2 (x - 4))) // Factor
8. Y^2 - (X^3 - 35*X + 98) /. {X ->
9.     1/7 (x + (
10.        112  (3 x^5 - x^4 - 210 x^3 - 1106 x^2 - 2625 x -
11.           3773))/(x^3 + 7 x^2 - 21 x - 91)^2),
12.    Y -> Sqrt[x^3 - 35*x - 98] /(
13.      7 Sqrt[7]) (1 - (
14.        112 (3 x^7 - 23 x^6 - 441 x^5 - 4571 x^4 - 33383 x^3 -
15.           135093 x^2 - 251811 x - 80409))/(x^3 + 7 x^2 - 21 x -
16.          91)^3)} // Factor

复制代码

\begin{align*}
&\qquad\qquad\qquad\>{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\raise{1pt}\tiny2(\sqrt{7+5\sqrt{2}}-2-\sqrt{2})(2+2\sqrt{2})^{1/4}}\>\!)^{2}t^2)}}\\
&\qquad\qquad=\frac{1}{2\sqrt{2}\,}{\Large\int}_{0}^{\chi(x,2\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\raise{1pt}\tiny{5+4\sqrt{2}-2\sqrt{14+10\sqrt{2}}}})^{2}t^2)}}\\
\\
&\chi(x,2\sqrt{2}\>\!)=\scriptsize{\tfrac{2\sqrt{2}x\sqrt{1-x^{2}}\big(1-2({\tiny{35+24\sqrt{2}-\sqrt{2254+1594\sqrt{2}}}})x^{2}+4({\tiny{392+277\sqrt{2}-2\sqrt{76574+54146\sqrt{2}}}})x^{4}-8({\tiny{488+345\sqrt{2}-2\sqrt{118999+84145\sqrt{2}}}})x^{6}\big)}
{\sqrt{1-4({\tiny{\sqrt{1598+1130\sqrt{2}}-28-20\sqrt{2}}})x^{2}}\big(1-2({\tiny{\sqrt{1054+746\sqrt{2}}-22-16\sqrt{2}}})x^{2}+4({\tiny{222+157\sqrt{2}-4\sqrt{6146+4346\sqrt{2}}}})x^{4}-16({\tiny{\sqrt{16921+11965\sqrt{2}}-92-65\sqrt{2}}})x^{6}\big)}}
\end{align*}

1. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-(2(1+\sqrt{2})(\sqrt{1+\sqrt{2}}-\sqrt{2})(2+2\sqrt{2})^{1/4})^{2}t^{2})}}dt
2. \frac{1}{2\sqrt{2}}\int_{0}^{\chi(x,2\sqrt{2})}\frac{1}{\sqrt{(1-t^{2})(1-(5+4\sqrt{2}-2(2+\sqrt{2})\sqrt{1+\sqrt{2}})^{2}t^{2})}}dt
3. \chi(x,s)=\left(\frac{1-2\left(35+24\sqrt{2}-\sqrt{2254+1594\sqrt{2}}\right)x^{2}+4\left(392+277\sqrt{2}-2\sqrt{76574+54146\sqrt{2}}\right)x^{4}-8\left(488+345\sqrt{2}-2\sqrt{118999+84145\sqrt{2}}\right)x^{6}}{1-2\left(\sqrt{1054+746\sqrt{2}}-16\sqrt{2}-22\right)x^{2}+4\left(222+157\sqrt{2}-4\sqrt{6146+4346\sqrt{2}}\right)x^{4}-16\left(\sqrt{16921+11965\sqrt{2}}-92-65\sqrt{2}\right)x^{6}}\right)\frac{2\sqrt{2}x\sqrt{1-x^{2}}}{\sqrt{1-(2(1+\sqrt{2})(\sqrt{1+\sqrt{2}}-\sqrt{2})(2+2\sqrt{2})^{1/4})^{2}x^{2}}}
4. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-(5+4\sqrt{2}-2(2+\sqrt{2})\sqrt{1+\sqrt{2}})^{2}t^{2})}}dt
5. \frac{1}{2\sqrt{2}}\int_{0}^{g}\frac{1}{\sqrt{(1+t^{2})(1+(5+4\sqrt{2}-2(2+\sqrt{2})\sqrt{1+\sqrt{2}})^{2}t^{2})}}dt
6. g=\left(\frac{1+(67+48\sqrt{2}-2\sqrt{2254+1594\sqrt{2}})x^{2}+(1431+1012\sqrt{2}-4\sqrt{255998+181018\sqrt{2}})x^{4}+(2405+1700\sqrt{2}-2\sqrt{2891006+2044250\sqrt{2}})x^{6}}{1-(47+32\sqrt{2}-2\sqrt{1054+746\sqrt{2}})x^{2}+(979+692\sqrt{2}-4\sqrt{119758+84682\sqrt{2}})x^{4}-(2405+1700\sqrt{2}-2\sqrt{2891006+2044250\sqrt{2}})x^{6}}\right)\frac{2\sqrt{2}x}{\sqrt{(1-x^{2})(1-(5+4\sqrt{2}-2\sqrt{14+10\sqrt{2}})^{2}x^{2})}}

复制代码

青青子衿

青青子衿 发表于 2023-8-24 12:26
\begin{align*}
&\qquad{\Large\int}_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\chi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{2}-1)^{2}t^{2}\big)}}\\
\\
&\qquad\>\>\>\chi(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\,x\sqrt{(1-x^{2})(1-2(\sqrt{2}-1)x^{2})}}{1-2(\sqrt{2}-1)x^{2}}\\
\\
\end{align*}

\begin{align*}
&\qquad{\Large\int}_{0}^{x}\sqrt{\frac{1-(\sqrt{2}-1)^{2}t^{2}}{1-t^{2}}}\>\!\mathrm{d}t\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\rho(x,\sqrt{2}\>\!)}\sqrt{\frac{1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}}{1-t^{2}}}\>\!\mathrm{d}t\\
&\qquad+\frac{2-\sqrt{2}}{2}{\Large\int}_{0}^{\rho(x,\sqrt{2}\>\!)
}
\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}\big)}}\\
&\qquad\qquad-\frac{2-\sqrt{2}}{2}\rho(x,\sqrt{2}\>\!)\sqrt{\big(1-x^{2}\big)\big(1-(\!\sqrt{2}-1)^{2}x^{2}\big)}\\
\\
&\qquad\qquad\qquad\quad\rho(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\,x}{1+(\sqrt{2}-1)x^{2}}\\
\\
&\qquad{\Large\int}_{0}^{x}\sqrt{\frac{1-(\!\sqrt{\small{2(\sqrt{2}-1)}}\>\>\!\!)^{2}t^{2}}{1-t^{2}}}\>\!\mathrm{d}t\\
&=\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\chi(x,\sqrt{2}\>\!)}\sqrt{\frac{1-(\sqrt{2}-1)^{2}t^{2}}{1-t^{2}}}\>\!\mathrm{d}t\\
&\qquad-\frac{2-\sqrt{2}}{2}{\Large\int}_{0}^{\chi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1-(\!\sqrt{2}-1)^{2}t^{2}\big)}}\\
&\qquad\qquad+\frac{2-\sqrt{2}}{2}\chi(x,\sqrt{2}\>\!)\\
\\
&\qquad\qquad\chi(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\,x\sqrt{(1-x^{2})(1-2(\sqrt{2}-1)x^{2})}}{1-2(\sqrt{2}-1)x^{2}}\\
\\
&\qquad{\Large\int}_{0}^{x}\sqrt{\frac{1+(\!\sqrt{\small{2(1+\sqrt{2})}}\>\>\!\!)^{2}t^{2}}{1-t^{2}}}\>\!\mathrm{d}t\\
&=-\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\varphi(x,\sqrt{2}\>\!)}\sqrt{\frac{1-(\!\sqrt{\small{2(1+\sqrt{2})}}\>\>\!\!)^{2}t^{2}}{1+t^{2}}}\>\!\mathrm{d}t\\
&\qquad+(1+\sqrt{2}\>\>\!\!){\Large\int}_{0}^{\varphi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1+t^{2}\big)\big(1-(\!\sqrt{\small{2(1+\sqrt{2})}}\>\>\!\!)^{2}t^{2}\big)}}\\
&\qquad\qquad-\varphi(x,\sqrt{2}\>\!)\sqrt{1+2(1+\sqrt{2}\>\!)x^{2}}\\
\\
&\qquad\qquad\qquad\quad\>\varphi(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\>\!x\sqrt{1-x^{2}}}{1+\sqrt{2}\>\!x^{2}}\\
\\
&\qquad{\Large\int}_{0}^{x}\sqrt{\frac{1-(\!\sqrt{\small{2(1+\sqrt{2})}}\>\>\!\!)^{2}t^{2}}{1+t^{2}}}\>\!\mathrm{d}t\\
&=-\frac{1}{\sqrt{2}\,}{\Large\int}_{0}^{\psi(x,\sqrt{2}\>\!)}\sqrt{\frac{1+(\!\sqrt{\small{2(1+\sqrt{2})}}\>\>\!\!)^{2}t^{2}}{1-t^{2}}}\>\!\mathrm{d}t\\
&\qquad+(1+\sqrt{2}\>\>\!\!){\Large\int}_{0}^{\psi(x,\sqrt{2}\>\!)}\frac{\mathrm{d}t}{\sqrt{\big(1-t^{2}\big)\big(1+(\!\sqrt{\small{2(1+\sqrt{2})}}\>\>\!\!)^{2}t^{2}\big)}}\\
&\qquad\qquad-\psi(x,\sqrt{2}\>\!)\sqrt{1-2(1+\sqrt{2}\>\!)x^{2}}\\
\\
&\qquad\qquad\qquad\quad\>\psi(x,\sqrt{2}\>\!)=\tfrac{\sqrt{2}\>\!x\sqrt{1+x^{2}}}{1-\sqrt{2}\>\!x^{2}}\\
\\

\end{align*}

青青子衿

\begin{align*}
\frac{1}{\sqrt{2}}\int_{0}^{\frac{\sqrt{2}x}{\small\sqrt{1+x^2}}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\small\frac{1}{2}}t^{2})}}
&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}\\
\\
\frac{1}{2}\int_{0}^{\frac{x\sqrt{1-x^{4}}}{1+x^{2}}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-(\sqrt{2}-1)^{2}t^{2})}}\qquad&\\
+\frac{1}{2\sqrt{2}}\int_{0}^{\frac{(1+\sqrt{2})x\sqrt{2(1-x^{8})}}{(1+x^{4})(1+\sqrt{2}-x^{2})}}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-(\sqrt{2}-1)^{2}t^{2})}}
&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1-t^{8}}}\\
\left(|x|\leqslant\tfrac{\sqrt{2\sqrt{10+8\sqrt{2}}-4-2\sqrt{2}}}{2}\right)\qquad\qquad
\end{align*}

青青子衿

\begin{gather*}
y^2=x^3-30x+56\\
\\
\left\{
\begin{split}
X&=\tfrac{1}{(\sqrt{2}\,i)^2}\left(x+\tfrac{18}{x-4}\right)\\
Y&=\tfrac{y}{(\sqrt{2}\,i)^3}\left(1-\tfrac{18}{(x-4)^2}\right)
\end{split}
\right.\\
\\
\begin{split}
(\varphi\circ\varphi\circ\mathcal{P})\oplus(2\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi^2+2)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-30 X+56\\
\end{gather*}

\begin{gather*}
y^2=x^3-35x+98\\
\\
\left\{
\begin{split}
\left.X\middle/\tfrac{1}{(\sqrt{7}\,i)^2}\right.&=x+\tfrac{
\begin{subarray}{l}
&112 (3x^5+x^4-210x^3\\
&\quad+1106x^2-2625 x+3773)
\end{subarray}
}{(x^3-7x^2-21x+91)^2}\\
\left.Y\middle/\tfrac{y}{(\sqrt{7}\,i)^3}\right.&=1-\tfrac{
\begin{subarray}{l}
&112 (3x^7+23x^6-441x^5\\
&\>\>+4571x^4-33383 x^3\\
&\quad+135093x^2-251811x\\
&\quad\>\>+80409)
\end{subarray}
}{(x^3-7 x^2-21x+91)^3}
\end{split}
\right.\\
\\
\begin{split}
(\varphi\circ\varphi\circ\mathcal{P})\oplus(7\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi^2+7)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-35 X+98
\end{gather*}


\begin{gather*}
y^2=x^3-35x+98\\
\\
\left\{
\begin{split}
X&=\tfrac{1}{\big(\frac{1+i \sqrt{7}}{2}\big)^2}\left(x-\tfrac{7+21 i \sqrt{7}}{2 x-7+i \sqrt{7}}\right)\\
Y&=\tfrac{y}{\big(\frac{1+i \sqrt{7}}{2}\big)^3}\left(1-\tfrac{2(7+21 i \sqrt{7})}{(2 x-7+i \sqrt{7}\,)^2}\right)
\end{split}
\right.\\
\\
\begin{split}
(\varphi\circ\varphi\circ\mathcal{P})\ominus(\varphi\circ\mathcal{P})\oplus(2\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi^2-\varphi+2)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-35 X+98
\end{gather*}



\begin{gather*}
y^2=x^3-264x+1694\\
\\
\left\{
\begin{split}
X&=\tfrac{1}{(\frac{1+i\sqrt{11}}{2})^{2}}\left(x+\tfrac{132(1-i\sqrt{11})x-88(11-14i\sqrt{11})}{(x-(11-i\sqrt{11}))^{2}}\right)\\
Y&=\tfrac{y}{(\frac{1+i\sqrt{11}}{2})^{3}}\left(1-\tfrac{132(1-i\sqrt{11})x-176(11-5i\sqrt{11})}{(x-(11-i\sqrt{11}))^{3}}\right)\\
\end{split}
\right.\\
\\
\begin{split}
(\varphi\circ\varphi\circ\mathcal{P})\ominus(\varphi\circ\mathcal{P})\oplus(3\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi^2-\varphi+3)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-264 X+1694
\end{gather*}

\begin{gather*}
y^2=x^3-264x+1694\\
\\
\left\{
\begin{split}
\left.X\middle/\tfrac{1}{(\sqrt{11}\,i)^2}\right.&=x+\tfrac{\begin{subarray}{l}
&792(8x^{9}-297x^{8}+792x^{7}+121440x^{6}-2509056x^{5}\\
&\quad+24480720x^{4}-173775360x^{3}+1195983360x^{2}\\
&\qquad-6093381888x+13564359424)
\end{subarray}}{(x^{5}-44x^{4}+220x^{3}+6776x^{2}-71632x+166496)^{2}}\\
\left.Y\middle/\tfrac{y}{(\sqrt{11}\,i)^3}\right.&=1-\tfrac{\begin{subarray}{l}
&1584(4x^{13}-121x^{12}-1452x^{11}+155276x^{10}\\
&\>\>-5672480x^{9}+161172000x^{8}-3521762112x^{7}\\
&\quad+54722116416x^{6}-572886762624x^{5}\\
&\quad\>\>+3741656771072x^{4}-12010526257664x^{3}\\
&\quad-9581302781952x^{2}+202938319108096x\\
&-464380338847744)
\end{subarray}}{(x^{5}-44x^{4}+220x^{3}+6776x^{2}-71632x+166496)^{3}}
\end{split}
\right.\\
\\
\begin{split}
(\varphi\circ\varphi\circ\mathcal{P})\oplus(11\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi^2+11)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-264 X+1694
\end{gather*}

青青子衿

【二次代数数复乘】

\begin{gather*}
{\Large{\int}}_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3-i \sqrt{7}}{8}}\,t)^{2})}}\\
\\
=\frac{1}{\frac{1+i \sqrt{7}}{2}}{\Large{\int}}_{0}^{\small\frac{4(1+i\sqrt{7})x}{8+(-3+i\sqrt{7})x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3-i \sqrt{7}}{8}}\,t)^{2})}}
\end{gather*}



\begin{gather*}
{\Large{\int}}_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3-i \sqrt{7}}{8}}\>\!t)^{2})}}\\
\\
=\frac{1}{\frac{\sqrt{7}+i}{2}}{\Large{\int}}_{0}^{\small\frac{(\sqrt{7}+i)x \sqrt{64-2(1-3 i \sqrt{7}) x^2}}{16+4(1+i \sqrt{7})x^2}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3 \sqrt{7}+i}{8}}\>\!t)^{2})}}\\
\\
{\Large{\int}}_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3 \sqrt{7}+i}{8}}\>\!t)^{2})}}\\
\\
=\frac{1}{\frac{-\sqrt{7}+i}{2}}{\Large{\int}}_{0}^{\small\frac{4(-\sqrt{7}+i)x\sqrt{1-x^{2}}}{8-(5+i\sqrt{7})x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3-i \sqrt{7}}{8}}\>\!t)^{2})}}\\
\end{gather*}



\begin{gather*}
\frac{{\Large{\int}}_{0}^{\small\frac{-7 \sqrt{21}+5 i \sqrt{3}}{46} }\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3-i \sqrt{7}}{8}}\>\!t)^{2})}}}{
{\Large{\int}}_{0}^{\small\frac{1}{2}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3 \sqrt{7}+i}{8}}\>\!t)^{2})}}}
=\frac{-\sqrt{7}+i}{2}\\
\end{gather*}

1. NIntegrate[1/Sqrt[(1 - x^2) (1 - ((3 - I Sqrt[7])/8)^2 x^2)], {x,
2.   0, (-7 Sqrt[21] + 5 I Sqrt[3])/
3.   46}]/NIntegrate[1/Sqrt[(1 -
4.     x^2) (1 - ((3 Sqrt[7] + I)/8)^2 x^2)], {x, 0, 1/2}]

复制代码