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高次有理椭圆模变换

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青青子衿 发表于 2023-7-31 10:01 |阅读模式
本帖最后由 青青子衿 于 2023-12-11 11:30 编辑
\begin{align*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}&=\frac{1}{2\sqrt{\scriptsize{k\sqrt{\left.k\middle/l\right.\,}\,}}+1}\int_{0}^{y_{3,1}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-l^{2}t^{2}\right)}}\\
\\
y_{3,1}&=\frac{x\left(1+2\sqrt{\scriptsize{k\sqrt{\left.k\middle/l\right.\,}\,}}+{\small{k\sqrt{\left.k\middle/l\right.\,}\,}}x^{2}\right)}{1+2\sqrt{\scriptsize{k\sqrt{\left.k\middle/l\right.\,}\,}}x^{2}+{\small{k\sqrt{\left.k\middle/l\right.\,}\,}}x^{2}}\\
\\
\int_{0}^{y_{3,2}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k'^{2}t^{2}\right)}}&=\left(2\sqrt{\scriptsize{l'\sqrt{\left.l'\middle/k'\right.\,}\,}}+1\right)\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-l'^{2}t^{2}\right)}}\\
\\
y_{3,2}&=\frac{x\left(1+2\sqrt{\scriptsize{l'\sqrt{\left.l'\middle/k'\right.\,}\,}}+{\small{l'\sqrt{\left.l'\middle/k'\right.\,}\,}}x^{2}\right)}{1+2\sqrt{\scriptsize{l'\sqrt{\left.l'\middle/k'\right.\,}\,}}x^{2}+{\small{l'\sqrt{\left.l'\middle/k'\right.\,}\,}}x^{2}}\\
\\
\frac{K\big(\sqrt{1-k^{2}}\,\big)}{K(k)}&=3\frac{K\big(\sqrt{1-l^{2}}\,\big)}{K(l)}\\
\\
3\frac{K\big(\sqrt{1-k'^{2}}\,\big)}{K(k')}&=\frac{K\big(\sqrt{1-l'^{2}}\,\big)}{K(l')}\\
\\
\frac{K\big(\sqrt{1-((2-\sqrt{3}\,)(\sqrt{3}-\sqrt{2}))^{2}}\,\big)}{K\big((2-\sqrt{3}\,)(\sqrt{3}-\sqrt{2}\,)\big)}
&=3\frac{K\big(\sqrt{1-((2-\sqrt{3}\,)(\sqrt{2}+\sqrt{3}))^{2}}\,\big)}{K\big((2-\sqrt{3}\,)(\sqrt{2}+\sqrt{3}\,)\big)}
=\sqrt{6}
\end{align*}
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 楼主| 青青子衿 发表于 2023-8-13 19:09
本帖最后由 青青子衿 于 2023-8-13 22:45 编辑
\begin{align*}
&\qquad\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-(2-\sqrt{3}\>\>\!\!)^{2}(\sqrt{3}-\sqrt{2}\>\>\!\!) ^2t^{2}\big)}}\\
&={\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)}}\\
&\qquad\left(3\sqrt{2}+2\sqrt{3}-3-\sqrt{6}\right){\Large\int}_{0}^{x}\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)}}\\
&={\Large\int}_{0}^{\frac{x(3\sqrt{2}+2\sqrt{3}-3-\sqrt{6}+(13-9\sqrt{2}-7\sqrt{3}+5\sqrt{6})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(10\sqrt{3}+7\sqrt{6}-17-12\sqrt{2}\>\>\!\!)}}\>\>\!\!)^{2}t^{2}\big)}}\\

\end{align*}



  1. \int_{0}^{x}\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
  2. *****
  3. \frac{1}{3+3\sqrt{2}-2\sqrt{3}-\sqrt{6}}\int_{0}^{\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}}}\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
  4. *****
  5. \int_{0}^{\frac{x\left(3\sqrt{2}+2\sqrt{3}-3-\sqrt{6}+\left(13-9\sqrt{2}-7\sqrt{3}+5\sqrt{6}\right)x^{2}\right)}{1+\left(9-6\sqrt{2}-5\sqrt{3}+4\sqrt{6}\right)x^{2}}}\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
  6. *****
  7. \left(3\sqrt{2}+2\sqrt{3}-3-\sqrt{6}\right)\int_{0}^{x}\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
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 楼主| 青青子衿 发表于 2023-12-10 11:51
本帖最后由 青青子衿 于 2025-1-31 20:08 编辑
青青子衿 发表于 2023-8-13 19:09
\begin{align*}
\color{black}{\frac{K\big(\sqrt{1-k^{2}}\,\big)}{K(k)}=3\frac{K\big(\sqrt{1-l^{2}}\,\big)}{K(l)}}\\
\\
\end{align*}



\begin{gather*}
I_3=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}=\frac{1}{2k^{3/4}l^{-1/4}+1}\int_{0}^{y_{3}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-l^{2}t^{2}\right)}}\\
\\
y_{3}=\frac{x\left(1+2k^{3/4}l^{-1/4}+k^{3/2}l^{-1/2}x^{2}\right)}{1+2k^{3/4}l^{-1/4}x^{2}+k^{3/2}l^{-1/2}x^{2}}\\
\\
k=\left(\frac{s^{3}\left(2+s\right)}{1+2s}\right)^{\frac{1}{2}}\qquad\,l=\left(s\left(\frac{2+s}{1+2s}\right)^{3}\right)^{\frac{1}{2}}\\
\\
0=\left(1-k^{2}\right)\left(1-l^{2}\right)-(1-k^{1/2}l^{1/2})^4\\
\end{gather*}

\begin{align*}
\kappa\Big(\frac{\tau}{2}\Big)&=\frac{2\sqrt{k(\tau)}}{1+k(\tau)}\\
l^{\frac{1}{4}}\Big(\frac{\tau}{3}\Big)&=\frac{\small\sqrt{k \sqrt{k}+\sqrt[3]{4 \sqrt{k} \left(1-k^2\right)}}-\sqrt{k\sqrt{k}}}{2}\\

&\qquad+\frac{1}{2} \sqrt{\scriptsize2 k \sqrt{k}-\sqrt[3]{4 \sqrt{k} \left(1-k^2\right)}+2 \sqrt{k^3-k\cdot\sqrt[3]{4 k^2 \left(1-k^2\right)}+2\cdot\sqrt[3]{2 k \left(1-k^2\right)^2}}}
\\
\end{align*}




  1. s=0.778
  2. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}\left(2+s\right)}{1+2s}t^{2}\right)}}dt
  3. \frac{1}{1+2s}\int_{0}^{\frac{x\left(1+2s+s^{2}x^{2}\right)}{1+2sx^{2}+s^{2}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-s\left(\frac{2+s}{1+2s}\right)^{3}t^{2}\right)}}dt
  4. k=\left(\frac{s^{3}\left(2+s\right)}{1+2s}\right)^{\frac{1}{2}}
  5. l=\left(s\left(\frac{2+s}{1+2s}\right)^{3}\right)^{\frac{1}{2}}
  6. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
  7. \frac{1}{2k^{3/4}l^{-1/4}+1}\int_{0}^{\frac{x\left(1+2k^{3/4}l^{-1/4}+k^{3/2}l^{-1/2}x^{2}\right)}{1+2k^{3/4}l^{-1/4}x^{2}+k^{3/2}l^{-1/2}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-l^{2}t^{2}\right)}}dt
  8. l^{\frac{1}{4}}-\frac{\sqrt{k^{3/2}+\sqrt[3]{4\sqrt{k}\left(1-k^{2}\right)}}-k^{3/4}+\sqrt{2k^{3/2}-\sqrt[3]{4\sqrt{k}\left(1-k^{2}\right)}+2\sqrt{k^{3}-\sqrt[3]{4k^{5}\left(1-k^{2}\right)}+\sqrt[3]{16k\left(1-k^{2}\right)^{2}}}}}{2}
  9. y^{\frac{1}{4}}=\frac{\sqrt{x^{\frac{3}{2}}+\sqrt[3]{4\sqrt{x}\left(1-x^{2}\right)}}-x^{\frac{3}{4}}+\sqrt{2x^{\frac{3}{2}}-\sqrt[3]{4\sqrt{x}\left(1-x^{2}\right)}+2\sqrt{x^{3}-\sqrt[3]{4x^{5}\left(1-x^{2}\right)}+\sqrt[3]{16x\left(1-x^{2}\right)^{2}}}}}{2}
  10. x^{\frac{1}{2}}y^{\frac{1}{2}}+\left(1-x^{2}\right)^{\frac{1}{4}}\left(1-y^{2}\right)^{\frac{1}{4}}=1
  11. u=\sqrt[4]{k}
  12. v=\frac{\sqrt{u^{6}+\sqrt[3]{4u^{2}\left(1-u^{8}\right)}}-u^{3}+\sqrt{2u^{6}-\sqrt[3]{4u^{2}\left(1-u^{8}\right)}+2\sqrt{u^{12}-\sqrt[3]{4u^{20}\left(1-u^{8}\right)}+\sqrt[3]{16u^{4}\left(1-u^{8}\right)^{2}}}}}{2}
  13. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
  14. \frac{v}{2u^{3}+v}\int_{0}^{\frac{x\left(v^{2}+2u^{3}v+u^{6}x^{2}\right)}{v^{2}+2u^{3}vx^{2}+u^{6}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}dt
  15. \left(1-u^{2}v^{2}\right)^{4}-\left(1-u^{8}\right)\left(1-v^{8}\right)
  16. N[16*k^2 + 24*k*Λ^2 + 48*k^2*Λ^2 +
  17.    24*k^3*Λ^2 + Λ^4 +
  18.    4*k*Λ^4 + 6*k^2*Λ^4 +
  19.    4*k^3*Λ^4 + k^4*Λ^4 -
  20.    8*Sqrt[k]*(1 + k)*Λ*(4 - 4*k + 4*k^2 -
  21.       3*Λ^2 + 10*k*Λ^2 -
  22.       3*k^2*Λ^2) /. {k -> ModularLambda[Sqrt[7] I]^(
  23.     1/2), Λ -> ModularLambda[(Sqrt[7] I)/6]^(1/2)}, 20]
复制代码



\begin{gather*}
I_3=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}=\frac{v}{2u^{3}+v}\int_{0}^{y_{3}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}\\
\\
y_{3}=\frac{x\left(v^{2}+2u^{3}v+u^{6}x^{2}\right)}{v^{2}+2u^{3}vx^{2}+u^{6}x^{2}}\\
\\
\begin{split}
u=\left(\frac{s^{3}\left(2+s\right)}{1+2s}\right)^{\frac{1}{8}}\qquad
v=\left(s\left(\frac{2+s}{1+2s}\right)^{3}\right)^{\frac{1}{8}}\\
\end{split}\\
\\
\\
\frac{1-y_{3}}{1+y_{3}}=\frac{1-x}{1+x}\left(\frac{1-sx}{1+sx}\right)^{2}\qquad
y_3=\frac{x\left({\small1+2s}+s^2x^{2}\right)}
{1+2sx^{2}+s^2x^{2}}
\end{gather*}


\begin{align*}
v&=\frac{\sqrt{u^{6}+\sqrt[3]{4u^{2}\left(1-u^{8}\right)}}-u^{3}}{2}\\
&\qquad\quad+\frac{\scriptsize{\sqrt{2u^{6}-\sqrt[3]{4u^{2}\left(1-u^{8}\right)}+2\sqrt{u^{12}-\sqrt[3]{4u^{20}\left(1-u^{8}\right)}+\sqrt[3]{16u^{4}\left(1-u^{8}\right)^{2}}}}}}{2}
\end{align*}



  1. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
  2. \frac{v}{2u^{3}+v}\int_{0}^{\frac{x\left(v^{2}+2u^{3}v+u^{6}x^{2}\right)}{v^{2}+2u^{3}vx^{2}+u^{6}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}dt
  3. \frac{1}{1+2s}\int_{0}^{\frac{x\left(1+2s+s^{2}x^{2}\right)}{1+2sx^{2}+s^{2}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}dt
  4. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}dt
  5. \frac{u}{u-2v^{3}}\int_{0}^{\frac{x\left(u^{2}-2uv^{3}+v^{6}x^{2}\right)}{u^{2}-2uv^{3}x^{2}+v^{6}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
  6. -\frac{1+2s}{3}\int_{0}^{\frac{x\left(1-2\left(\frac{2+s}{1+2s}\right)+\frac{(2+s)^{2}}{(1+2s)^{2}}x^{2}\right)}{1-2\left(\frac{2+s}{1+2s}\right)x^{2}+\frac{(2+s)^{2}}{(1+2s)^{2}}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
  7. s=0.589
  8. u=\left(\frac{s^{3}\left(2+s\right)}{1+2s}\right)^{1/8}
  9. v=\left(s\left(\frac{2+s}{1+2s}\right)^{3}\right)^{1/8}
  10. uv^{3}-\left(\frac{s(2+s)}{(1+2s)}\right)\left(\frac{(2+s)^{1/4}}{s^{1/4}(1+2s)^{1/4}}\right)
  11. u^{3}v-\frac{s(2+s)}{(1+2s)}\left(\frac{s^{1/4}(1+2s)^{1/4}}{(2+s)^{1/4}}\right)
  12. v^{2}-\left(\frac{2+s}{1+2s}\right)\left(\frac{s^{1/4}(1+2s)^{1/4}}{(2+s)^{1/4}}\right)
  13. u^{2}-s\left(\frac{(2+s)^{1/4}}{s^{1/4}(1+2s)^{1/4}}\right)
  14. v^{6}-\frac{s(2+s)^{2}}{(1+2s)^{2}}\left(\frac{(2+s)^{1/4}}{s^{1/4}(1+2s)^{1/4}}\right)
  15. u^{6}-\frac{s^{2}(2+s)}{(1+2s)}\left(\frac{s^{1/4}(1+2s)^{1/4}}{(2+s)^{1/4}}\right)
  16. u^{5}v^{3}-\frac{s^{2}(2+s)^{2}}{(1+2s)^{2}}\left(\frac{s^{1/4}(1+2s)^{1/4}}{(2+s)^{1/4}}\right)
  17. u^{3}v^{5}-\frac{s^{2}(2+s)^{2}}{(1+2s)^{2}}\left(\frac{(2+s)^{1/4}}{s^{1/4}(1+2s)^{1/4}}\right)
  18. u^{2}v^{4}-\frac{s(2+s)^{2}}{(1+2s)^{2}}\left(\frac{s^{1/4}(1+2s)^{1/4}}{(2+s)^{1/4}}\right)
  19. u^{4}v^{2}-\frac{s^{2}(2+s)}{(1+2s)}\left(\frac{(2+s)^{1/4}}{s^{1/4}(1+2s)^{1/4}}\right)
复制代码



\begin{align*}
\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise1px\scriptsize\frac{s^{3}\left(2+s\right)}{1+2s}}t^{2}\right)}}

&=\frac{1}{1+2s}\int_{0}^{\frac{x\left(1+2s+s^{2}x^{2}\right)}{1+2sx^{2}+s^{2}x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise1px\scriptsize\frac{s(2+s)^3\,}{(1+2s)^{3}}}t^{2}\right)}}\\
\\
\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise1px\scriptsize\frac{s(2+s)^3\,}{(1+2s)^{3}}}t^{2}\right)}}

&=-\frac{1+2s}{3}\int_{0}^{\frac{x\left(1-2\left(\frac{2+s}{1+2s}\right)+\frac{(2+s)^{2}}{(1+2s)^{2}}x^{2}\right)}{1-2\left(\frac{2+s}{1+2s}\right)x^{2}+\frac{(2+s)^{2}}{(1+2s)^{2}}x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise1px\scriptsize\frac{s^{3}\left(2+s\right)}{1+2s}}t^{2}\right)}}
\end{align*}

\begin{align*}
s\rightarrow-\frac{2+s}{1+2s}
\end{align*}



  1. s=0.947
  2. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}\left(2+s\right)}{1+2s}t^{2}\right)}}dt
  3. \frac{1}{1+2s}\int_{0}^{\frac{x\left(1+2s+s^{2}x^{2}\right)}{1+2sx^{2}+s^{2}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-s\left(\frac{2+s}{1+2s}\right)^{3}t^{2}\right)}}dt
  4. \frac{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(1-s)(1+s)^{3}}{1+2s}t^{2}\right)}}dt}{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}\left(2+s\right)}{1+2s}t^{2}\right)}}dt}
  5. 3\frac{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2}\right)}}dt}{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-s\left(\frac{2+s}{1+2s}\right)^{3}t^{2}\right)}}dt}
  6. u=\frac{2^{2/3}3^{1/2}-3^{1/2}-1}{2}
  7. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{u\left(2+u\right)^{3}}{\left(1+2u\right)^{3}}t^{2}\right)}}dt
  8. -\frac{1+2u}{3}\int_{0}^{\frac{x\left(1-2\frac{2+u}{1+2u}+\frac{(2+u)^{2}}{(1+2u)^{2}}x^{2}\right)}{1-2\frac{2+u}{1+2u}x^{2}+\frac{(2+u)^{2}}{(1+2u)^{2}}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{u^{3}\left(2+u\right)}{1+2u}t^{2}\right)}}dt
  9. \frac{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(1-u)^{3}(1+u)}{(1+2u)^{3}}t^{2}\right)}}dt}{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{u\left(2+u\right)^{3}}{\left(1+2u\right)^{3}}t^{2}\right)}}dt}
  10. \frac{1}{3}\cdot\frac{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(1-u)(1+u)^{3}}{1+2u}t^{2}\right)}}dt}{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{u^{3}\left(2+u\right)}{1+2u}t^{2}\right)}}dt}
  11. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-(\frac{\sqrt{6}-\sqrt{2}}{4})^{2}t^{2})}}dt
  12. \frac{1}{\sqrt{3}}\int_{0}^{\frac{\sqrt{3}x(1+\frac{2\sqrt{3}-3}{6}x^{2})}{1+\frac{\sqrt{3}}{2}x^{2}}}\frac{1}{\sqrt{(1-t^{2})(1-(\frac{\sqrt{6}+\sqrt{2}}{4})^{2}t^{2})}}dt
  13. -\frac{(1-2^{1/3}+2^{2/3})^{2}3^{1/2}}{9}\int_{0}^{\frac{x\left(-\frac{(1+2^{1/3})^{2}3^{1/2}}{3}+\left(\frac{3+3^{1/2}+2^{2/3}3^{1/2}+2^{4/3}3^{1/2}}{6}\right)^{2}x^{2}\right)}{1-\left(\frac{2^{1/6}3^{3/4}+2^{1/2}3^{3/4}-2^{1/6}3^{5/4}}{6}\right)^{2}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-(\frac{2^{7/6}3+2^{17/6}3^{1/2}-2^{1/2}3^{2}-2^{7/6}3^{1/2}-2^{1/2}3^{1/2}}{12})^{2}t^{2}\right)}}dt
  14. -\frac{(1-2^{1/3}+2^{2/3})^{2}3^{1/2}}{9}\int_{0}^{\frac{-\frac{(1+2^{1/3})^{2}3^{1/2}}{3}x\left(1-\left(\frac{3^{1/4}-2^{1/3}3^{1/4}+2^{2/3}3^{1/4}+3^{3/4}+2^{1/3}3^{3/4}}{6}\right)^{2}x^{2}\right)}{1-\left(\frac{2^{1/6}3^{3/4}+2^{1/2}3^{3/4}-2^{1/6}3^{5/4}}{6}\right)^{2}x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-(\frac{2^{7/6}3+2^{17/6}3^{1/2}-2^{1/2}3^{2}-2^{7/6}3^{1/2}-2^{1/2}3^{1/2}}{12})^{2}t^{2}\right)}}dt
复制代码



\begin{align*}
\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise{0.7px}\scriptsize(\frac{\sqrt{6}-\sqrt{2}}{4})^{2}}t^{2}\right)}}

&=\frac{1}{\sqrt{3}}\int_{0}^{\frac{\sqrt{3}x(1+{\scriptsize\frac{2\sqrt{3}-3}{6}}x^{2})}{1+{\scriptsize\frac{\sqrt{3}}{2}}x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise{0.7px}\scriptsize(\frac{\sqrt{6}+\sqrt{2}}{4})^{2}}t^{2}\right)}}\\
\end{align*}
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 楼主| 青青子衿 发表于 2023-12-11 11:59
本帖最后由 青青子衿 于 2024-3-2 22:33 编辑
青青子衿 发表于 2023-12-10 11:51
\begin{align*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}&=\frac{1}{2k^{3/4}l^{-1/4}+1}\int_{0}^{y_{3,1}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-l^{2}t^{2}\right)}}\\
\\
y_{3,1}&=\frac{x\left(1+2k^{3/4}l^{-1/4}+k^{3/2}l^{-1/2}x^{2}\right)}{1+2k^{3/4}l^{-1/4}x^{2}+k^{3/2}l^{-1/2}x^{2}}\\
\\
\end{align*}





\begin{gather*}
I_5=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}=\frac{v(1-uv^{3})}{v-u^{5}}\int_{0}^{y_{5}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}\\
\\
y_{5}=\frac{x(v(u^{5}-v)+u^{3}(u^{2}+v^{2})(u^{5}-v)x^{2}+u^{10}(uv^{3}-1)x^{4})}{v^{2}(uv^{3}-1)+uv^{2}(u^{2}+v^{2})(u^{5}-v)x^{2}+u^{6}v^{3}(u^{5}-v)x^{4}}\\
\\
\begin{split}
u&=\left(\frac{1}{2}-\frac{1+s-s^{2}}{2}\sqrt{\frac{1+s^{2}}{1+2s}}\,\right)^{\frac{1}{8}}\\
v&=\left(\frac{1}{2}-\frac{1-11s-s^{2}}{2(1+2s)^{2}}\sqrt{\frac{1+s^{2}}{1+2s}}\,\right)^{\frac{1}{8}}\\
\end{split}\\
\\
\begin{split}
\frac{v(1-uv^{3})}{v-u^{5}}&=\frac{1}{1+2s}\\
{\scriptsize\frac{u^{3}(u^{2}+v^{2})(u^{5}-v)}{v^{2}(uv^{3}-1)}}&={\scriptsize(1+s)\sqrt{(1+2s)(1+s^{2})}-1-2s}\\
{\scriptsize\frac{u^{10}}{v^{2}}}&={\scriptsize\frac{1+2s+s^{2}+s^{3}-(1+s)\sqrt{(1+2s)(1+s^{2})}}{2}}\\
{\scriptsize\frac{u(u^{2}+v^{2})(u^{5}-v)}{uv^{3}-1}}&={\scriptsize\sqrt{(1+2s)(1+s^{2})}-1+s+s^{2}}\\
{\scriptsize\frac{u^{6}v(u^{5}-v)}{uv^{3}-1}}&={\scriptsize\frac{1-s^{2}+s^{3}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2}}\\

\end{split}\\
\\

\end{gather*}

\begin{gather*}
\quad\begin{split}
0&=u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})\\
0&=(u^{2}-v^{2})(u^{4}+6u^{2}v^{2}+v^{4})+4uv(1-u^{4}v^{4})\\
0&=(u^{2}-v^{2})^{6}-16u^{2}v^{2}(1-u^{8})(1-v^{8})\\
1&=u^{4}v^{4}+((1-u^{8})(1-v^{8}))^{1/2}+2(16u^{8}v^{8}(1-u^{8})(1-v^{8}))^{1/6}
\end{split}\\
\\
\begin{split}
u^{8}+v^{8}&=1+\xi^{8}-\left(\frac{\sqrt[3]{9-9\xi^{4}+\sqrt{81+222\xi^{4}+81\xi^{8}}}}{\sqrt[3]{18}}\right.\\
&\qquad\qquad\left.-\frac{\sqrt[3]{9\xi^{4}-9+\sqrt{81+222\xi^{4}+81\xi^{8}}}}{\sqrt[3]{18}}\right)^{6}
\end{split}
\end{gather*}



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 楼主| 青青子衿 发表于 2023-12-12 01:04
本帖最后由 青青子衿 于 2024-4-2 04:38 编辑
青青子衿 发表于 2023-12-11 11:59
\begin{align*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}=\frac{v(uv^{3}-1)}{u^{5}-v}\int_{0}^{y_{\scriptsize5}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}
\end{align*}



\begin{gather*}
I_7=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-U_7t^{2}\right)}}
=M_{7}\int_{0}^{y_{\scriptsize7}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-V_7t^{2}\right)}}\\
\\
y_{\scriptsize7}=\frac{\frac{1}{M_7}x(1+\tau_{\scriptsize7,1}x^2+\tau_{\scriptsize7,2}x^4+\tau_{\scriptsize7,3}x^6)}{1+\omega_{\scriptsize7,1}x^2+\omega_{\scriptsize7,2}x^4+\omega_{\scriptsize7,3}x^6}\\
\\
\\
\qquad
\begin{split}
\delta_{\small7}&=\sqrt{\scriptsize(1-\xi+2\xi^{2})(2-\xi+\xi^{2})(2-3\xi+2\xi^{2})}\\
M_{\small7}&=\tfrac{3-8\xi+6\xi^{2}-4\xi^{3}+2\delta_7}{7}\\
\tfrac{1}{M_7}&=-{\scriptsize\,(3-8\xi+6\xi^{2}-4\xi^{3})+2\delta_7}\\
U_{\small7}&=\tfrac{1+\xi^{8}-(1-\xi)^{8}}{2}-{\scriptsize\,2\xi(1-\xi)(1-\xi+\xi^{2})\delta_7}\\
V_{\small7}&=\tfrac{1+\xi^{8}-(1-\xi)^{8}}{2}+{\scriptsize\,2\xi(1-\xi)(1-\xi+\xi^{2})\delta_7}\\
\end{split}\qquad\quad\\
\\
\qquad\>\begin{split}
\tau_{\scriptsize7,1}&={\scriptsize-(4-10\xi+17\xi^{2}-18\xi^{3}+10\xi^{4}-4\xi^{5})+(2-2\xi+2\xi^{2})\delta_{7}}\\

\tau_{\scriptsize7,2}&={\scriptsize8-32\xi+78\xi^{2}-126\xi^{3}+147\xi^{4}-126\xi^{5}+78\xi^{6}-32\xi^{7}+8\xi^{8}}\\
&
\quad\>\>\;{\scriptsize-\,(4-10\xi+16\xi^{2}-16\xi^{3}+10\xi^{4}-4\xi^{5})\delta_{7}}\\

\tau_{\scriptsize7,3}&={\small-\,\tfrac{32-160\xi+456\xi^{2}-864\xi^{3}+1176\xi^{4}-1176\xi^{5}+861\xi^{6}-448\xi^{7}+154\xi^{8}-28\xi^{9}}{7}}\\
&\qquad{\small+\,\tfrac{2(8-28\xi+56\xi^{2}-70\xi^{3}+56\xi^{4}-28\xi^{5}+7\xi^{6})\delta_{7}}{7}}\\
\omega_{\scriptsize7,1}&=-{\scriptsize\,\xi(1-2\xi)(2-\xi+2\xi^{2})(3-2\xi+2\xi^{2})+2\xi(2-\xi+2\xi^{2})\delta_7}\\
\omega_{\scriptsize7,2}&={\scriptsize\xi^{2}(28-126\xi+319\xi^{2}-534\xi^{3}+636\xi^{4}-536\xi^{5}+328\xi^{6}-128\xi^{7}+32\xi^{8})}\\
&\qquad-{\scriptsize\,2\xi^{2}(7-21\xi+34\xi^{2}-36\xi^{3}+20\xi^{4}-8\xi^{5})\delta_7}\\
\omega_{\scriptsize7,3}&=-{\scriptsize\xi^{3}(28-154\xi+448\xi^{2}-861\xi^{3}+1176\xi^{4})}\\
&\qquad+{\scriptsize\xi^{3}(1176\xi^{5}-864\xi^{6}+456\xi^{7}-160\xi^{8}+32\xi^{9})}\\
&\qquad\quad+{\scriptsize2\xi^{3}(7-28\xi+56\xi^{2}-70\xi^{3}+56\xi^{4}-28\xi^{5}+8\xi^{6})\delta_7}\\
\end{split}\\
\end{gather*}

  1. \xi=0.597
  2. \delta_{7}=\sqrt{(1-\xi+2\xi^{2})(2-\xi+\xi^{2})(2-3\xi+2\xi^{2})}
  3. M_{7}=\frac{3-8\xi+6\xi^{2}-4\xi^{3}+2\delta_{7}}{7}
  4. U_{7}=\frac{1+\xi^{8}-\left(1-\xi\right)^{8}}{2}-2\xi\left(1-\xi\right)\left(1-\xi+\xi^{2}\right)\delta_{7}
  5. V_{7}=\frac{1+\xi^{8}-\left(1-\xi\right)^{8}}{2}+2\xi\left(1-\xi\right)\left(1-\xi+\xi^{2}\right)\delta_{7}
  6. \tau_{71}=-(4-10\xi+17\xi^{2}-18\xi^{3}+10\xi^{4}-4\xi^{5})+(2-2\xi+2\xi^{2})\delta_{7}
  7. \tau_{72}=8-32\xi+78\xi^{2}-126\xi^{3}+147\xi^{4}-126\xi^{5}+78\xi^{6}-32\xi^{7}+8\xi^{8}-(4-10\xi+16\xi^{2}-16\xi^{3}+10\xi^{4}-4\xi^{5})\delta_{7}
  8. \tau_{73}=-\frac{32-160\xi+456\xi^{2}-864\xi^{3}+1176\xi^{4}-1176\xi^{5}+861\xi^{6}-448\xi^{7}+154\xi^{8}-28\xi^{9}}{7}+\frac{2(8-28\xi+56\xi^{2}-70\xi^{3}+56\xi^{4}-28\xi^{5}+7\xi^{6})\delta_{7}}{7}
  9. \omega_{71}=-\xi(1-2\xi)(2-\xi+2\xi^{2})(3-2\xi+2\xi^{2})+2\xi(2-\xi+2\xi^{2})\delta_{7}
  10. \omega_{72}=\xi^{2}(28-126\xi+319\xi^{2}-534\xi^{3}+636\xi^{4}-536\xi^{5}+328\xi^{6}-128\xi^{7}+32\xi^{8})-2\xi^{2}(7-21\xi+34\xi^{2}-36\xi^{3}+20\xi^{4}-8\xi^{5})\delta_{7}
  11. \omega_{73}=-\xi^{3}(28-154\xi+448\xi^{2}-861\xi^{3}+1176\xi^{4}-1176\xi^{5}+864\xi^{6}-456\xi^{7}+160\xi^{8}-32\xi^{9})+2\xi^{3}(7-28\xi+56\xi^{2}-70\xi^{3}+56\xi^{4}-28\xi^{5}+8\xi^{6})\delta_{7}
  12. y_{7}=\frac{\frac{1}{M_{7}}x\left(1+\tau_{71}x^{2}+\tau_{72}x^{4}+\tau_{73}x^{6}\right)}{1+\omega_{71}x^{2}+\omega_{72}x^{4}+\omega_{73}x^{6}}
  13. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-U_{7}t^{2})}}dt
  14. M_{7}\int_{0}^{y_{7}}\frac{1}{\sqrt{(1-t^{2})(1-V_{7}t^{2})}}dt
  15. g_{7}=2(1-\xi)(3-3\xi+2\xi^{2})
  16. h_{7}=g_{7}-7M_{7}+\frac{1}{M_{7}}
  17. 2\xi(2-\xi+2\xi^{2})
  18. \int_{0}^{x}\frac{g_{7}}{\sqrt{(1-t^{2})(1-U_{7}t^{2})}}dt-7M_{7}\int_{0}^{x}\sqrt{\frac{1-U_{7}t^{2}}{1-t^{2}}}dt+\int_{0}^{y_{7}}\sqrt{\frac{1-V_{7}t^{2}}{1-t^{2}}}dt
  19. \frac{h_{7}x\left(1+\sigma_{71}x^{2}+\sigma_{72}x^{4}\right)\sqrt{(1-x^{2})(1-U_{7}x^{2})}}{1+\omega_{71}x^{2}+\omega_{72}x^{4}+\omega_{73}x^{6}}
  20. \sigma_{71}=\frac{-2\xi(4-10\xi+17\xi^{2}-18\xi^{3}+10\xi^{4}-4\xi^{5})+4\xi(1-\xi+\xi^{2})\delta_{7}}{2-\xi+2\xi^{2}}
  21. \sigma_{72}=\frac{3\xi^{2}(4-14\xi+28\xi^{2}-35\xi^{3}+28\xi^{4}-14\xi^{5}+4\xi^{6})-6\xi^{2}(1-\xi)(1-\xi+\xi^{2})\delta_{7}}{2-\xi+2\xi^{2}}
  22. \int_{0}^{1}\frac{M_{7}\frac{Q_{7}}{\Gamma_{7}}}{(1-V_{7}\gamma_{7}^{2}t^{2})\sqrt{(1-t^{2})(1-V_{7}t^{2})}}dt
  23. \int_{0}^{1}\frac{7\Upsilon_{7}}{(1-U_{7}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_{7}t^{2})}}dt-\int_{0}^{1}\frac{6P_{7}}{\sqrt{(1-t^{2})(1-U_{7}t^{2})}}dt
  24. \beta=0.965
  25. \gamma_{7}=\frac{\frac{1}{M_{7}}\beta\Upsilon_{7}}{\Gamma_{7}}
  26. \Upsilon_{7}=1+\tau_{71}\beta^{2}+\tau_{72}\beta^{4}+\tau_{73}\beta^{6}
  27. \Gamma_{7}=1+\omega_{71}\beta^{2}+\omega_{72}\beta^{4}+\omega_{73}\beta^{6}
  28. P_{71}=\frac{4(1-\xi+\xi^{2})\delta_{7}-2(4-10\xi+17\xi^{2}-18\xi^{3}+10\xi^{4}-4\xi^{5})}{3}
  29. P_{72}=\frac{(2-\xi+2\xi^{2})(4-14\xi+28\xi^{2}-35\xi^{3}+28\xi^{4}-14\xi^{5}+4\xi^{6})-2(1-\xi)(1-\xi+\xi^{2})(2-\xi+2\xi^{2})\delta_{7}}{3}
  30. Q_{71}=2(1-\xi)(3-2\xi+2\xi^{2})\delta_{7}-2(6-18\xi)-2(35\xi^{2}-40\xi^{3}+30\xi^{4}-14\xi^{5}+4\xi^{6})
  31. Q_{72}=40-104\xi+26\xi^{2}+568\xi^{3}-1861\xi^{4}+3416\xi^{5}-4294\xi^{6}+3912\xi^{7}-2600\xi^{8}+1232\xi^{9}-384\xi^{10}+64\xi^{11}-2(1-\xi)(10-\xi-31\xi^{2}+95\xi^{3}-130\xi^{4})\delta_{7}-2(1-\xi)(112\xi^{5}-56\xi^{6}+16\xi^{7})\delta_{7}
  32. Q_{73}=4(1-\xi)(4+32\xi-178\xi^{2}+522\xi^{3}-1025\xi^{4})\delta_{7}+4(1-\xi)(1516\xi^{5}-1736\xi^{6}+1552\xi^{7}-1056\xi^{8})\delta_{7}+4(1-\xi)(528\xi^{9}-176\xi^{10}+32\xi^{11})\delta_{7}-4(29190\xi^{9}-21792\xi^{10}+13096\xi^{11})-4(8+44\xi-488\xi^{2}+2136\xi^{3}-6076\xi^{4})+4(6160\xi^{12}-2160\xi^{13}+512\xi^{14}-64\xi^{15})-4(12838\xi^{5}-21363\xi^{6}+28832\xi^{7}-31972\xi^{8})
  33. Q_{74}=\xi(320-1952\xi+5360\xi^{2}-4816\xi^{3}-19796\xi^{4})+\xi(102452\xi^{5}-269276\xi^{6}+503335\xi^{7}-729820\xi^{8})+\xi(850228\xi^{9}-807988\xi^{10}+628880\xi^{11})-\xi(399024\xi^{12}-203552\xi^{13}+81280\xi^{14})+\xi(24192\xi^{15}-4864\xi^{16}+512\xi^{17})-4\xi(1-\xi)(40-144\xi+170\xi^{2}+437\xi^{3})\delta_{7}+4\xi(1-\xi)(2403\xi^{4}-5845\xi^{5}+9571\xi^{6})\delta_{7}-4\xi(1-\xi)(11623\xi^{7}-10806\xi^{8}+7744\xi^{9})\delta_{7}+4\xi(1-\xi)(4216\xi^{10}-1680\xi^{11}+448\xi^{12}-64\xi^{13})\delta_{7}
  34. Q_{75}=2\xi^{2}(1-\xi)(192-1456\xi+5992\xi^{2}-16832\xi^{3})\delta_{7}+2\xi^{2}(1-\xi)(35392\xi^{4}-58268\xi^{5}+76975\xi^{6})\delta_{7}+2\xi^{2}(1-\xi)(-82614\xi^{7}+72338\xi^{8}-51496\xi^{9})\delta_{7}+2\xi^{2}(1-\xi)(+29456\xi^{10}-13216\xi^{11}+4448\xi^{12})\delta_{7}+2\xi^{2}(1-\xi)(-1024\xi^{13}+128\xi^{14})\delta_{7}-2\xi^{2}(384-3872\xi+20608\xi^{2}-74872\xi^{3})-2\xi^{2}(205184\xi^{4}-446460\xi^{5}+794682\xi^{6})+2\xi^{2}(1178478\xi^{7}-1472331\xi^{8}+1559180\xi^{9})-2\xi^{2}(1402450\xi^{10}-1069642\xi^{11}+687764\xi^{12})+2\xi^{2}(368784\xi^{13}-161952\xi^{14}+56544\xi^{15})-2\xi^{2}(14912\xi^{16}-2688\xi^{17}+256\xi^{18})
  35. Q_{76}=\xi^{3}(4-14\xi+28\xi^{2}-35\xi^{3}+28\xi^{4}-14\xi^{5}+4\xi^{6})(64-448\xi+1680\xi^{2}-4256\xi^{3}+7952\xi^{4}-11424\xi^{5}+12865\xi^{6}-11424\xi^{7}+7952\xi^{8}-4256\xi^{9}+1680\xi^{10}-448\xi^{11}+64\xi^{12})-2\xi^{3}(1-\xi)(1-\xi+\xi^{2})(8-28\xi+56\xi^{2}-71\xi^{3}+56\xi^{4}-28\xi^{5}+8\xi^{6})(8-28\xi+56\xi^{2}-69\xi^{3}+56\xi^{4}-28\xi^{5}+8\xi^{6})\delta_{7}
  36. P_{7}=1+P_{71}\beta^{2}+P_{72}\beta^{4}
  37. Q_{7}=1+Q_{71}\beta^{2}+Q_{72}\beta^{4}+Q_{73}\beta^{6}+Q_{74}\beta^{8}+Q_{75}\beta^{10}+Q_{76}\beta^{12}
复制代码


\begin{gather*}
I_{7}=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-U_7t^{2}\right)}}
=M_7\int_{0}^{y_{7}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-V_7t^{2}\right)}}\\
\\
\frac{1-y_{7}}{1+y_{7}}=\frac{1-x}{1+x}\left(\frac{1-\alpha_{7,1}x+\alpha_{7,2}x^{2}-\alpha_{7,3}x^{3}}{1+\alpha_{7,1}x+\alpha_{7,2}x^{2}+\alpha_{7,3}x^{3}}\right)^2\\
\\
\\
\begin{split}
\delta_{7}&=\sqrt{\scriptsize(1-\xi+2\xi^{2})(2-\xi+\xi^{2})(2-3\xi+2\xi^{2})}\\
M_7&=\tfrac{3-8\xi+6\xi^{2}-4\xi^{3}}{7}+\tfrac{2\delta_7}{7}\\
U_7&=\tfrac{1+\xi^{8}-(1-\xi)^{8}}{2}-{\scriptsize\,2\xi(1-\xi)(1-\xi+\xi^{2})\delta_7}
\\
V_7&=\tfrac{1+\xi^{8}-(1-\xi)^{8}}{2}+{\scriptsize\,2\xi(1-\xi)(1-\xi+\xi^{2})\delta_7}
\\
\alpha_{7,1}&=-{\scriptsize\,(2-4\xi+3\xi^{2}-2\xi^{3})+\delta_7}\\
\alpha_{7,2}&=-{\scriptsize\,(2-7\xi+14\xi^{2}-16\xi^{3}+10\xi^{4}-4\xi^{5})}\\
&\qquad+{\scriptsize\,(1-2 \xi +2 \xi ^2)\delta_7}\\
\alpha_{7,3}&={\scriptsize\,4-14 \xi+28 \xi ^2-35 \xi ^3+28 \xi ^4-14 \xi ^5+4 \xi ^6}\\
&\qquad
-{\scriptsize\,2(1-\xi) (1-\xi +\xi ^2)\delta_7}
\end{split}

\end{gather*}




  1. \xi=0.769
  2. u=\left(R-2S\delta_{7}\right)^{\frac{1}{8}}
  3. v=\left(R+2S\delta_{7}\right)^{\frac{1}{8}}
  4. R=\frac{1+\xi^{8}-\left(1-\xi\right)^{8}}{2}
  5. S=\xi\left(1-\xi\right)\left(1-\xi+\xi^{2}\right)
  6. T=\left(1-\xi+2\xi^{2}\right)\left(2-\xi+\xi^{2}\right)\left(2-3\xi+2\xi^{2}\right)
  7. u\cdot v-\xi
  8. y_{7}\left(x,\varphi\right)=\frac{\frac{1}{M_{7}}x\left(1+\tau_{1}x^{2}+\tau_{2}x^{4}+\tau_{3}x^{6}\right)}{1+\omega_{1}x^{2}+\omega_{2}x^{4}+\omega_{3}x^{6}}
  9. \alpha_{1}=\frac{1}{2}\left(\frac{1}{M_{7}}-1\right)
  10. \alpha_{2}=\frac{1}{2}\left(\frac{u^{3}v^{3}}{M_{7}}-\frac{u^{7}}{v}\right)
  11. \alpha_{3}=\frac{u^{7}}{v}
  12. \tau_{1}=\left(\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}+2\alpha_{2}+2\alpha_{3}\right)M_{7}
  13. \tau_{2}=\left(\alpha_{2}^{2}+2\alpha_{1}\alpha_{3}+2\alpha_{2}\alpha_{3}\right)M_{7}
  14. \tau_{3}=\alpha_{3}^{2}M_{7}
  15. \omega_{1}=\alpha_{1}^{2}+2\alpha_{1}+2\alpha_{2}
  16. \omega_{2}=\alpha_{2}^{2}+2\alpha_{1}\alpha_{2}+2\alpha_{1}\alpha_{3}+2\alpha_{3}
  17. \omega_{3}=\alpha_{3}^{2}+2\alpha_{2}\alpha_{3}
  18. M_{7}=\frac{v\left(1-uv\right)\left(1-uv+u^{2}v^{2}\right)}{v-u^{7}}
  19. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
  20. M_{7}\int_{0}^{y_{7}\left(x,1\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}dt
  21. \left(28-126\xi+319\xi^{2}-534\xi^{3}+636\xi^{4}-536\xi^{5}+328\xi^{6}-128\xi^{7}+32\xi^{8}-2(7-21\xi+34\xi^{2}-36\xi^{3}+20\xi^{4}-8\xi^{5})\delta_{7}\right)M_{7}
  22. -(4-10\xi+17\xi^{2}-18\xi^{3}+10\xi^{4}-4\xi^{5})+(2-2\xi+2\xi^{2})\delta_{7}
  23. \left(-(2-\xi+2\xi^{2})(28-154\xi+448\xi^{2}-861\xi^{3}+1176\xi^{4}-1176\xi^{5}+864\xi^{6}-456\xi^{7}+160\xi^{8}-32\xi^{9})+2(2-\xi+2\xi^{2})(7-28\xi+56\xi^{2}-70\xi^{3}+56\xi^{4}-28\xi^{5}+8\xi^{6})\delta_{7}\right)M_{7}
  24. 8-32\xi+78\xi^{2}-126\xi^{3}+147\xi^{4}-126\xi^{5}+78\xi^{6}-32\xi^{7}+8\xi^{8}-(4-10\xi+16\xi^{2}-16\xi^{3}+10\xi^{4}-4\xi^{5})\delta_{7}
  25. \left(32-224\xi+840\xi^{2}-2128\xi^{3}+3976\xi^{4}-5712\xi^{5}+6433\xi^{6}-5712\xi^{7}+3976\xi^{8}-2128\xi^{9}+840\xi^{10}-224\xi^{11}+32\xi^{12}-4(1-\xi)(1-\xi+\xi^{2})(4-14\xi+28\xi^{2}-35\xi^{3}+28\xi^{4}-14\xi^{5}+4\xi^{6})\delta_{7}\right)M_{7}
  26. -\frac{32-160\xi+456\xi^{2}-864\xi^{3}+1176\xi^{4}-1176\xi^{5}+861\xi^{6}-448\xi^{7}+154\xi^{8}-28\xi^{9}}{7}+\frac{2(8-28\xi+56\xi^{2}-70\xi^{3}+56\xi^{4}-28\xi^{5}+7\xi^{6})\delta_{7}}{7}
  27. \delta_{7}=\sqrt{(1-\xi+2\xi^{2})(2-\xi+\xi^{2})(2-3\xi+2\xi^{2})}
复制代码



\begin{align*}
0&=\left(1-u^{8}\right)\left(1-v^{8}\right)-\left(1-uv\right)^{8}\\
\\
uv&=\xi\\
\sqrt{\tfrac{u^{7}}{v}}&=(1-\xi+\xi^{2})\sqrt{\small2-3\xi+2\xi^{2}}\\
&\qquad-(1-\xi)\sqrt{\small(2-\xi+\xi^{2})(1-\xi+2\xi^{2})}\\
\left(\tfrac{v^{8}\left(1-v^{8}\right)}{u^{8}\left(1-u^{8}\right)}\right)^{1/6}
&=\frac{2-7\xi+11\xi^{2}-8\xi^{3}+4\xi^{4}}{2\xi(1-\xi)}\\
&\qquad+\frac{1-2\xi}{2\xi(1-\xi)}\sqrt{\small\left(1-\xi+2\xi^{2}\right)\left(2-\xi+\xi^{2}\right)\left(2-3\xi+2\xi^{2}\right)}\\
\left(\tfrac{u^{8}\left(1-u^{8}\right)}{v^{8}\left(1-v^{8}\right)}\right)^{1/6}
&=\frac{2-7\xi+11\xi^{2}-8\xi^{3}+4\xi^{4}}{2\xi(1-\xi)}\\
&\qquad\>-\frac{1-2\xi}{2\xi(1-\xi)}\sqrt{\small\left(1-\xi+2\xi^{2}\right)\left(2-\xi+\xi^{2}\right)\left(2-3\xi+2\xi^{2}\right)}\\
\\
M&=\tfrac{v(1-uv)\left(1-uv+u^{2}v^{2}\right)}{v-u^{7}}\\
\alpha_1&=\tfrac{1}{2}\left(\tfrac{1}{M}-1\right)\\
\alpha_2&=\tfrac{1}{2}\left(\tfrac{u^{3}v^{3}}{M}-\tfrac{u^{7}}{v}\right)\\
\alpha_3&=\tfrac{u^{7}}{v}\\
\end{align*}
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 楼主| 青青子衿 发表于 2023-12-15 08:36
本帖最后由 青青子衿 于 2024-1-9 11:58 编辑
青青子衿 发表于 2023-12-12 01:04
\begin{align*}
I_3=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}&=\frac{v}{2u^{3}+v}\int_{0}^{y_{3,1}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-v^{8}t^{2}\right)}}\\
\\
y_{3,1}&=\frac{x\left(v^{2}+2u^{3}v+u^{6}x^{2}\right)}{v^{2}+2u^{3}vx^{2}+u^{6}x^{2}}\\
\\
\\
\end{align*}



\begin{align*}
\Phi(x,U)&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}\\
\Phi(y,V)&=\int_{0}^{y}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-Vt^{2}\right)}}\\
\mathcal{E}(x,U)&=\int_{0}^{x}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}\mathrm{d}t\\
\mathcal{E}(y,V)&=\int_{0}^{y}\sqrt{\frac{1-Vt^{2}}{1-t^{2}}}\mathrm{d}t\\
U&=u^8\\
V&=v^8\\
y&=\frac{x \left(v^2+2 u^3 v+u^6 x^2\right)}{v^2+2 u^3 v x^2+u^6 x^2}\\
\frac{\partial\,y}{\partial\,x}&=\frac{(v^{2}-u^{6}x^{2})(v^{2}+2u^{3}v-2u^{3}vx^{2}-u^{6}x^{2})}{(v^{2}+2u^{3}vx^{2}+u^{6}x^{2})^{2}}\\
\frac{\partial\,y}{\partial\,u}&=\frac{6u^{2}vx(1-x^{2})(v^{2}-u^{6}x^{2})}{(v^{2}+2u^{3}vx^{2}+u^{6}x^{2})^{2}}\\

\frac{\partial\,y}{\partial\,v}&=-\frac{2u^{3}x\ (1-x^{2})(v^{2}-u^{6}x^{2})}{(v^{2}+2u^{3}vx^{2}+u^{6}x^{2})^{2}}=-\dfrac{u}{3v}\cdot\frac{\partial\,y}{\partial\,u}\\
M(u,v)&=\dfrac{v}{2u^3+v}=\dfrac{2v^3-u}{3u}\\
N(u,v)&=\frac{1}{M(u,v)}=\dfrac{3u}{2v^3-u}=\dfrac{2u^3+v}{v}\\
\\

\\
\Phi(x,U)&=M(u,v)\Phi(y,V)\\
\Phi(y,V)&=\frac{\Phi(x,U)}{M(u,v)}=N(u,v)\Phi(x,U)\\
\frac{\partial\,\Phi(x,U)}{\partial\,U}&=\frac{\mathcal{E}(x,U)}{2U\left(1-U\right)}-\dfrac{\Phi(x,U)}{2U}\\
&\qquad\quad-\frac{x(1-x^2)}{2(1-U)}\cdot\frac{\partial\,\Phi(x,U)}{\partial\,x}\\
\frac{\partial\,\Phi(y,V)}{\partial\,V}
&=\frac{\mathcal{E}(y,V)}{2V\left(1-V\right)}-\dfrac{\Phi(y,V)}{2V}\\
&\qquad\quad-\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,\Phi(y,V)}{\partial\,y}\\

\\
\Psi(u,v)&=(1-u^{2}v^{2})^4-\left(1-u^{8}\right)\left(1-v^{8}\right)\\
\frac{\mathrm{d}v}{\mathrm{d}u}&=-\dfrac{\Psi_u(u,v)}{\Psi_v(u,v)}=\frac{v(1-v^{8})}{3u(1-u^{8})M^{2}(u,v)}\\
&=\frac{(1-v^{8})(2u^3+v)^2}{3uv(1-u^{8})}\\
\\
\end{align*}




\begin{align*}
\frac{\partial\,\Phi(y,V)}{\partial\,u}&=\frac{\partial\,\Phi(y,V)}{\partial\,V}\cdot\frac{\partial\,V}{\partial\,u}+\frac{\partial\,\Phi(y,V)}{\partial\,y}\cdot\left(\frac{\partial\,y}{\partial\,u}+\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}\right)\\
&=\left(-\dfrac{\frac{\partial\,V}{\partial\,u}}{2V}\right)\cdot\Phi(y,V)+\frac{\frac{\partial\,V}{\partial\,u}}{2V\left(1-V\right)}\cdot\mathcal{E}(y,V)\\

&\quad\>\>+\frac{\partial\,\Phi(y,V)}{\partial\,y}\cdot\left(\frac{\partial\,y}{\partial\,u}+\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}-\frac{y(1-y^2)\frac{\partial\,V}{\partial\,u}}{2(1-V)}\right)\\

&=\left(-\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}\right)\cdot\Phi(x,U)+\frac{\frac{\partial\,V}{\partial\,u}}{2V\left(1-V\right)}\cdot\mathcal{E}(y,V)\\
&\qquad\quad+\dfrac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M(u,v)}\cdot\frac{\frac{\partial\,y}{\partial\,u}+\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}-\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,V}{\partial\,u}}{\frac{\partial\,\!y}{\partial\,\!x}}\\
\\
\frac{\partial}{\partial\,u}\left(\frac{\Phi(x,U)}{M(u,v)}\right)&=\left(\frac{\partial\,\!N(u,v)}{\partial\,u}+\frac{\partial\,\!N(u,v)}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}\right)\cdot\Phi(x,U)\\
&\qquad\quad+\frac{\frac{\partial\,U}{\partial\,u}}{M(u,v)}\cdot\frac{\partial\,\Phi(x,U)}{\partial\,U}\\
&=\left(N_0(u,v)-\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}\right)\cdot\Phi(x,U)\\
&\qquad\quad+\frac{\frac{\partial\,U}{\partial\,u}}{2U\left(1-U\right)M(u,v)}\cdot\mathcal{E}(x,U)\\
&\qquad\qquad\quad-\frac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M(u,v)}\cdot\frac{x(1-x^2)\frac{\partial\,U}{\partial\,u}}{2(1-U)}\\

\end{align*}

\begin{align*}
\Omega&=\frac{\frac{\partial\,V}{\partial\,u}}{2V\left(1-V\right)}\cdot\mathcal{E}(y,V)-\frac{\frac{\partial\,U}{\partial\,u}}{2U\left(1-U\right)M(u,v)}\cdot\mathcal{E}(x,U)\\
&\qquad\quad-\left(N_0(u,v)-\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}+\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}\right)\cdot\Phi(x,U)\\
\\
&=\dfrac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M(u,v)}\cdot\left(\frac{\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,\!V}{\partial\,\!u}-\frac{\partial\,y}{\partial\,u}-\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}}{\frac{\partial\,\!y}{\partial\,\!x}}-\frac{x(1-x^2)\cdot\frac{\partial\,U}{\partial\,u}}{2(1-U)}\right)\\
\\
\end{align*}



\begin{align*}
&\\
\\
\frac{\partial\,V}{\partial\,u}&=\frac{\partial\,V}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}=8v^7\cdot\frac{(1-v^{8})(2u^3+v)^2}{3uv(1-u^{8})}\\
&=\frac{8v^6(1-v^{8})(2u^{3}+v)^{2}}{3u(1-u^{8})}\\
\frac{\partial\,\Phi(y,V)}{\partial\,y}&=\dfrac{1}{\sqrt{(1-y^2)(1-Vy^2)}}=\dfrac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M(u,v)\frac{\partial\,\!y}{\partial\,\!x}}\\

\frac{\partial\,y}{\partial\,u}+\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}&=\left(1-\dfrac{u}{3v}\cdot\frac{(1-v^{8})(2u^3+v)^2}{3uv(1-u^{8})}\right)\cdot\frac{\partial\,y}{\partial\,u}\\
&=\left(1-\frac{\left(1-v^{8}\right)(2u^{3}+v)^{2}}{9v^2\left(1-u^{8}\right)}\right)\cdot\frac{\partial\,y}{\partial\,u}=\frac{vN_0}{6u^2}\cdot\frac{\partial\,y}{\partial\,u}\\
\\
N_0&=\frac{\partial\,N(u,v)}{\partial\,u}+\frac{\partial\,N(u,v)}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}\\
&=\frac{6 u^2}{v}\cdot\left(1-\frac{\left(1-v^8\right) \left(2 u^3+v\right)^2}{9v^2\left(1-u^8\right) }\right)\\



\end{align*}








  1. s=0.56
  2. F\left(\rho,\mu\right)=\int_{0}^{\rho}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\mu t^{2}\right)}}dt
  3. E\left(\rho,\mu\right)=\int_{0}^{\rho}\sqrt{\frac{1-\mu t^{2}}{1-t^{2}}}dt
  4. u=x
  5. U=u^{8}
  6. \frac{d}{dx}\left(F\left(s,U\right)\right)
  7. \left(\frac{E\left(s,U\right)}{2U\left(1-U\right)}-\frac{F\left(s,U\right)}{2U}-\frac{s\left(1-s^{2}\right)}{2\left(1-U\right)}\frac{1}{\sqrt{\left(1-s^{2}\right)\left(1-Us^{2}\right)}}\right)\frac{d}{dx}\left(U\right)
  8. M=\frac{1}{1+u^{4}}
  9. r=\frac{\left(1+u^{4}\right)s}{1+u^{4}s^{2}}
  10. V=v^{8}
  11. \frac{F\left(s,U\right)}{M}
  12. F\left(r,V\right)
  13. \frac{d}{dx}\left(\frac{F\left(s,U\right)}{M}\right)
  14. \left(\frac{d}{dx}\left(\frac{1}{M}\right)\right)F\left(s,U\right)+\frac{8u^{7}}{M}\left(\frac{E\left(s,U\right)}{2U\left(1-U\right)}-\frac{F\left(s,U\right)}{2U}-\frac{s\left(1-s^{2}\right)}{2\left(1-U\right)}\frac{1}{\sqrt{\left(1-s^{2}\right)\left(1-Us^{2}\right)}}\right)
  15. \frac{d}{dx}\left(F\left(r,V\right)\right)
  16. \left(\frac{E\left(r,V\right)}{2V\left(1-V\right)}-\frac{F\left(r,V\right)}{2V}-\frac{r\left(1-r^{2}\right)}{2\left(1-V\right)}\frac{1}{\sqrt{\left(1-r^{2}\right)\left(1-Vr^{2}\right)}}\right)\left(\frac{d}{dx}\left(V\right)\right)+\frac{\frac{d}{dx}\left(r\right)}{\sqrt{\left(1-r^{2}\right)\left(1-Vr^{2}\right)}}
  17. v=\left(\frac{2u^{2}}{1+u^{4}}\right)^{\frac{1}{4}}
  18. \frac{1+u^{4}}{2}
  19. \frac{8v^{7}\cdot\frac{v\left(1-v^{8}\right)}{2u\left(1-u^{8}\right)M^{2}}\cdot\frac{1}{2v^{8}\left(1-v^{8}\right)}}{\frac{8u^{7}}{M}\cdot\frac{1}{2u^{8}\left(1-u^{8}\right)}}
  20. \frac{\left(\frac{d}{dx}\left(V\right)\right)E\left(r,V\right)}{2V\left(1-V\right)}-\frac{8u^{7}E\left(s,U\right)}{2U\left(1-U\right)M}-\left(\left(\frac{d}{dx}\left(\frac{1}{M}\right)\right)-\frac{8u^{7}}{2UM}+\frac{\left(\frac{d}{dx}\left(V\right)\right)}{2VM}\right)F\left(s,U\right)
  21. \frac{\frac{r\left(1-r^{2}\right)}{2\left(1-V\right)}\left(\frac{d}{dx}\left(V\right)\right)-\left(\frac{d}{dx}\left(r\right)\right)}{\sqrt{\left(1-r^{2}\right)\left(1-Vr^{2}\right)}}-\frac{\frac{8u^{7}s\left(1-s^{2}\right)}{2M\left(1-U\right)}}{\sqrt{\left(1-s^{2}\right)\left(1-Us^{2}\right)}}
  22. \left(\frac{\frac{r\left(1-r^{2}\right)}{2\left(1-V\right)}\left(\frac{d}{dx}\left(V\right)\right)-\left(\frac{d}{dx}\left(r\right)\right)}{G}-\frac{8u^{7}s\left(1-s^{2}\right)}{2\left(1-U\right)}\right)\cdot\frac{1}{M\sqrt{\left(1-s^{2}\right)\left(1-Us^{2}\right)}}
  23. f\left(p,a,b\right)=\frac{\left(1+a^{4}\right)p}{1+a^{4}p^{2}}
  24. g\left(x,a,b\right)=\frac{d}{dx}\left(f\left(x,a,b\right)\right)
  25. G=g\left(s,u,v\right)
  26. \frac{1}{\sqrt{\left(1-r^{2}\right)\left(1-Vr^{2}\right)}}
  27. \frac{1}{GM\sqrt{\left(1-s^{2}\right)\left(1-Us^{2}\right)}}
  28. E\left(r,V\right)-2M\cdot E\left(s,U\right)-\frac{2V\left(1-V\right)}{\frac{d}{dx}\left(V\right)}\left(\left(\frac{d}{dx}\left(\frac{1}{M}\right)\right)-\frac{8u^{7}}{2UM}+\frac{\left(\frac{d}{dx}\left(V\right)\right)}{2VM}\right)F\left(s,U\right)
  29. \frac{2u\left(1-u^{8}\right)M}{4\sqrt{\left(1-s^{2}\right)\left(1-Us^{2}\right)}}\left(\frac{\frac{r\left(1-r^{2}\right)}{2\left(1-V\right)}\left(\frac{d}{dx}\left(V\right)\right)-\left(\frac{d}{dx}\left(r\right)\right)}{G}-\frac{8u^{7}s\left(1-s^{2}\right)}{2\left(1-U\right)}\right)
复制代码



\begin{align*}
\Omega^*&=\frac{\Omega}{\frac{\frac{\partial\,V}{\partial\,u}}{2V\left(1-V\right)}}\\
&=\mathcal{E}(y,V)-\frac{\frac{V\left(1-V\right)\frac{\partial\,U}{\partial\,u}}{U\left(1-U\right)\frac{\partial\,V}{\partial\,v}}}{\frac{v(1-v^{8})}{nu(1-u^{8})M^{2}(u,v)}M(u,v)}\cdot\mathcal{E}(x,U)\\
&\qquad\quad-\frac{2V\left(1-V\right)}{\frac{\partial\,V}{\partial\,u}}\left(N_0(u,v)-\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}+\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}\right)\cdot\Phi(x,U)\\
\\
&=\dfrac{2V\left(1-V\right)\cdot\frac{\partial\,\Phi(x,U)}{\partial\,x}}{\frac{\partial\,V}{\partial\,v}\cdot\frac{v(1-v^{8})}{nu(1-u^{8})M^{2}(u,v)}M(u,v)}\cdot\left(\frac{\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,\!V}{\partial\,\!u}-\frac{\partial\,y}{\partial\,u}-\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}}{\frac{\partial\,\!y}{\partial\,\!x}}-\frac{x(1-x^2)\cdot\frac{\partial\,U}{\partial\,u}}{2(1-U)}\right)\\
\\
\Omega^*&=\mathcal{E}(y,V)-nM(u,v)\cdot\mathcal{E}(x,U)\\
&\qquad-\frac{2V\left(1-V\right)}{\frac{\partial\,V}{\partial\,u}}\left(N_0(u,v)-\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}+\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}\right)\cdot\Phi(x,U)\\
\\
&=\dfrac{nu(1-u^{8})M(u,v)}{4\sqrt{(1-x^2)(1-u^8x^2)}}\cdot\left(\frac{\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,\!V}{\partial\,\!u}-\frac{\partial\,y}{\partial\,u}-\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}}{\frac{\partial\,\!y}{\partial\,\!x}}-\frac{x(1-x^2)\cdot\frac{\partial\,U}{\partial\,u}}{2(1-U)}\right)\\
\end{align*}


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 楼主| 青青子衿 发表于 2023-12-31 22:59
本帖最后由 青青子衿 于 2024-2-9 22:12 编辑
青青子衿 发表于 2023-12-15 08:36
\begin{align*}
\Omega&=\frac{\frac{\partial\,V}{\partial\,u}}{2V\left(1-V\right)}\cdot\mathcal{E}(y,V)-\frac{\frac{\partial\,U}{\partial\,u}}{2U\left(1-U\right)M(u,v)}\cdot\mathcal{E}(x,U)\\
&\qquad\quad-\left(N_0(u,v)-\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}+\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}\right)\cdot\Phi(x,U)\\
\\
&=\dfrac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M(u,v)}\cdot\left(\frac{\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,\!V}{\partial\,\!u}-\frac{\partial\,y}{\partial\,u}-\frac{\partial\,y}{\partial\,v}\cdot\frac{\mathrm{d}v}{\mathrm{d}u}}{\frac{\partial\,\!y}{\partial\,\!x}}-\frac{x(1-x^2)\cdot\frac{\partial\,U}{\partial\,u}}{2(1-U)}\right)\\
\\
\end{align*}



\begin{align*}
&\qquad\qquad\qquad\Psi(u,v)\>\colon\!=(1-u^{2}v^{2})^4-\left(1-u^{8}\right)\left(1-v^{8}\right)\equiv0\\
&\qquad\qquad\qquad\qquad\>\>\>\>r=\frac{s\left(v^{2}+2u^{3}v+u^{6}s^{2}\right)}{v^{2}+2u^{3}vs^{2}+u^{6}s^{2}}\\
\\
\Omega&=\frac{4\left(2u^{3}+v\right)^{2}}{3uv^{2}\left(1-u^{8}\right)}\int_{0}^{r}\sqrt{\frac{1-v^{8}t^{2}}{1-t^{2}}}\mathrm{d}t-\frac{4\left(2u^{3}+v\right)}{uv\left(1-u^{8}\right)}\int_{0}^{s}\sqrt{\frac{1-u^{8}t^{2}}{1-t^{2}}}\mathrm{d}t\\

&\qquad-\left(\frac{2\left(1-v^{8}\right)\left(3u^{3}+2v\right)\left(2u^{3}+v\right)^{2}}{3uv^{3}\left(1-u^{8}\right)}-\frac{2\left(u^{3}+2v\right)}{uv}\right)\int_{0}^{s}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}\\
&=\frac{6u^{2}\left(2u^{3}+v\right)^{3}s}{(1-u^{8})\sqrt{\frac{1-u^{8}s^{2}}{1-s^2}}}\left(\frac{\frac{2v^{5}\left(v^{2}-u^{6}s^{2}\right)\left(v^{2}+2u^{3}v+u^{6}s^{2}\right)}{9u^{3}\left(v^{2}+2u^{3}vs^{2}+u^{6}s^{2}\right)}+\frac{1-v^{8}}{9v^{2}}-\frac{1-u^{8}}{\left(2u^{3}+v\right)^{2}}}{v^{2}+2u^{3}v-2u^{3}vs^{2}-u^{6}s^{2}}-\frac{2u^{5}}{3v\left(2u^{3}+v\right)^{2}}\right)\\

\end{align*}

\begin{align*}
\\
\frac{\Omega}{\Omega_1}&=\int_{0}^{r}\sqrt{\frac{1-v^{8}t^{2}}{1-t^{2}}}\mathrm{d}t-\frac{3v}{2u^{3}+v}\int_{0}^{s}\sqrt{\frac{1-u^{8}t^{2}}{1-t^{2}}}\mathrm{d}t\\

&\qquad-\left(\frac{\left(1-v^{8}\right)\left(3u^{3}+2v\right)}{2v}-\frac{3v(1-u^{8})\left(u^{3}+2v\right)}{4(2u^{3}+v)^2}\right)\int_{0}^{s}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}\\
&=\frac{9u^{3}v^2\left(2u^{3}+v\right)s}{2\sqrt{\frac{1-u^{8}s^{2}}{1-s^2}}}\left(\frac{\frac{2v^{5}\left(v^{2}-u^{6}s^{2}\right)\left(v^{2}+2u^{3}v+u^{6}s^{2}\right)}{9u^{3}\left(v^{2}+2u^{3}vs^{2}+u^{6}s^{2}\right)}+\frac{1-v^{8}}{9v^{2}}-\frac{1-u^{8}}{\left(2u^{3}+v\right)^{2}}}{v^{2}+2u^{3}v-2u^{3}vs^{2}-u^{6}s^{2}}-\frac{2u^{5}}{3v\left(2u^{3}+v\right)^{2}}\right)\\


\end{align*}

\begin{align*}
\Omega&=\Omega_1\cdot\mathcal{E}(y,V)-\Omega_2\cdot\mathcal{E}(x,U)-\Omega_3\cdot\Phi(x,U)\\
&=\Omega_4\cdot\dfrac{1}{M(u,v)}\cdot\frac{\partial\,\Phi(x,U)}{\partial\,x}\\
\Omega_1&=\frac{\frac{\partial\,V}{\partial\,u}}{2V\left(1-V\right)}
=\frac{8v^6(1-v^{8})(2u^{3}+v)^{2}}{2v^8(1-v^8)\cdot3u(1-u^{8})}\\
&=\frac{4(2u^{3}+v)^{2}}{3uv^2(1-u^{8})}\\
\Omega_2&=\frac{\frac{\partial\,U}{\partial\,u}}{2U\left(1-U\right)M(u,v)}=\frac{8u^7(2u^3+v)}{2u^8(1-u^8)v}\\
&=\frac{4(2u^3+v)}{uv(1-u^8)}\\
N_0&=\frac{6 u^2}{v}\cdot\left(1-\frac{\left(1-v^8\right) \left(2 u^3+v\right)^2}{9v^2\left(1-u^8\right) }\right)\\
\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}
&=\frac{8u^7(2u^3+v)}{2u^8v}=\frac{4(2u^3+v)}{uv}\\
\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}
&=\frac{2u^3+v}{2v^8v}\cdot\frac{8v^6(1-v^{8})(2u^{3}+v)^{2}}{3u(1-u^{8})}\\
&=\frac{4\left(1-v^{8}\right)\left(2u^{3}+v\right)^{3}}{3uv^{3}\left(1-u^{8}\right)}\\
\Omega_3&=N_0-\frac{\frac{\partial\,U}{\partial\,u}}{2UM(u,v)}+\dfrac{\frac{\partial\,V}{\partial\,u}}{2VM(u,v)}\\
&=\frac{2\left(1-v^{8}\right)\left(3u^{3}+2v\right)\left(2u^{3}+v\right)^{2}}{3uv^{3}\left(1-u^{8}\right)}\\
&\qquad\quad-\frac{2\left(u^{3}+2v\right)}{uv}
\end{align*}




  2. s=0.56
  3. F\left(\rho,\mu\right)=\int_{0}^{\rho}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\mu t^{2}\right)}}dt
  4. E\left(\rho,\mu\right)=\int_{0}^{\rho}\sqrt{\frac{1-\mu t^{2}}{1-t^{2}}}dt
  5. U\left(u,a\right)=u^{8}
  6. \frac{d}{dx}\left(F\left(s,U\left(x,a\right)\right)\right)
  7. \left(\frac{E\left(s,U\left(x,a\right)\right)}{2U\left(x,a\right)\left(1-U\left(x,a\right)\right)}-\frac{F\left(s,U\left(x,a\right)\right)}{2U\left(x,a\right)}-\frac{s\left(1-s^{2}\right)}{2\left(1-U\left(x,a\right)\right)}\frac{1}{\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}\right)\frac{d}{dx}\left(U\left(x,a\right)\right)
  8. M\left(u,a\right)=\frac{v\left(u,a\right)}{2u^{3}+v\left(u,a\right)}
  9. r\left(u,a\right)=\frac{s\left(v\left(u,a\right)^{2}+2u^{3}v\left(u,a\right)+u^{6}s^{2}\right)}{v\left(u,a\right)^{2}+2u^{3}v\left(u,a\right)s^{2}+u^{6}s^{2}}
  10. V\left(u,a\right)=v\left(u,a\right)^{8}
  11. \frac{F\left(s,U\left(x,a\right)\right)}{M\left(x,a\right)}
  12. F\left(r\left(x,a\right),V\left(x,a\right)\right)
  13. \frac{d}{dx}\left(\frac{F\left(s,U\left(x,a\right)\right)}{M\left(x,a\right)}\right)
  14. \left(\frac{d}{dx}\left(\frac{1}{M\left(x,a\right)}\right)\right)F\left(s,U\left(x,a\right)\right)+\frac{\frac{d}{dx}\left(U\left(x,a\right)\right)}{M\left(x,a\right)}\left(\frac{E\left(s,U\left(x,a\right)\right)}{2U\left(x,a\right)\left(1-U\left(x,a\right)\right)}-\frac{F\left(s,U\left(x,a\right)\right)}{2U\left(x,a\right)}-\frac{s\left(1-s^{2}\right)}{2\left(1-U\left(x,a\right)\right)}\frac{1}{\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}\right)
  15. \frac{d}{dx}\left(F\left(r\left(x,a\right),V\left(x,a\right)\right)\right)
  16. \left(\frac{E\left(r\left(x,a\right),V\left(x,a\right)\right)}{2V\left(x,a\right)\left(1-V\left(x,a\right)\right)}-\frac{F\left(r\left(x,a\right),V\left(x,a\right)\right)}{2V\left(x,a\right)}-\frac{r\left(x,a\right)\left(1-r\left(x,a\right)^{2}\right)}{2\left(1-V\left(x,a\right)\right)}\frac{1}{\sqrt{\left(1-r\left(x,a\right)^{2}\right)\left(1-V\left(x,a\right)r\left(x,a\right)^{2}\right)}}\right)\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)+\frac{\frac{d}{dx}\left(r\left(x,a\right)\right)}{\sqrt{\left(1-r\left(x,a\right)^{2}\right)\left(1-V\left(x,a\right)r\left(x,a\right)^{2}\right)}}
  17. v\left(u,a\right)=\frac{\sqrt{u^{6}+\sqrt[3]{4u^{2}\left(1-u^{8}\right)}}-u^{3}+\sqrt{2u^{6}-\sqrt[3]{4u^{2}\left(1-u^{8}\right)}+2\sqrt{u^{12}-\sqrt[3]{4u^{20}\left(1-u^{8}\right)}+\sqrt[3]{16u^{4}\left(1-u^{8}\right)^{2}}}}}{2}
  18. \frac{\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)E\left(r\left(x,a\right),V\left(x,a\right)\right)}{2V\left(x,a\right)\left(1-V\left(x,a\right)\right)}-\frac{\left(\frac{d}{dx}\left(U\left(x,a\right)\right)\right)E\left(s,U\left(x,a\right)\right)}{2U\left(x,a\right)\left(1-U\left(x,a\right)\right)M\left(x,a\right)}-\left(\left(\frac{d}{dx}\left(\frac{1}{M\left(x,a\right)}\right)\right)-\frac{\frac{d}{dx}\left(U\left(x,a\right)\right)}{2U\left(x,a\right)M\left(x,a\right)}+\frac{\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)}{2V\left(x,a\right)M\left(x,a\right)}\right)F\left(s,U\left(x,a\right)\right)
  19. \frac{\frac{r\left(x,a\right)\left(1-r\left(x,a\right)^{2}\right)}{2\left(1-V\left(x,a\right)\right)}\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)-\left(\frac{d}{dx}\left(r\left(x,a\right)\right)\right)}{\sqrt{\left(1-r\left(x,a\right)^{2}\right)\left(1-V\left(x,a\right)r\left(x,a\right)^{2}\right)}}-\frac{\left(\frac{d}{dx}\left(U\left(x,a\right)\right)\right)\frac{s\left(1-s^{2}\right)}{2M\left(x,a\right)\left(1-U\left(x,a\right)\right)}}{\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}
  20. \left(\frac{\frac{r\left(x,a\right)\left(1-r\left(x,a\right)^{2}\right)}{2\left(1-V\left(x,a\right)\right)}\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)-\left(\frac{d}{dx}\left(r\left(x,a\right)\right)\right)}{G\left(x,a\right)}-\frac{\left(\frac{d}{dx}\left(U\left(x,a\right)\right)\right)s\left(1-s^{2}\right)}{2\left(1-U\left(x,a\right)\right)}\right)\cdot\frac{1}{M\left(x,a\right)\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}
  21. \left(\frac{\frac{s\left(1-s^{2}\right)\left(v(x,a)^{2}+2x^{3}v(x,a)+x^{6}s^{2}\right)\left(v(x,a)^{2}-s^{2}x^{6}\right)^{2}}{\left(v(x,a)^{2}+2x^{3}v(x,a)s^{2}+x^{6}s^{2}\right)^{3}}\cdot\frac{4v(x,a)^{6}\left(1-v(x,a)^{8}\right)\left(2x^{3}+v(x,a)\right)^{2}}{3x\left(1-v(x,a)^{8}\right)\left(1-x^{8}\right)}-\left(1-\frac{\left(1-v(x,a)^{8}\right)\left(2x^{3}+v(x,a)\right)^{2}}{9v(x,a)^{2}\left(1-x^{8}\right)}\right)\frac{6x^{2}v(x,a)s(1-s^{2})(v(x,a)^{2}-x^{6}s^{2})}{(v(x,a)^{2}+2x^{3}v(x,a)s^{2}+x^{6}s^{2})^{2}}}{\frac{(v(x,a)^{2}-x^{6}s^{2})(v(x,a)^{2}+2x^{3}v(x,a)-2x^{3}v(x,a)s^{2}-x^{6}s^{2})}{(v(x,a)^{2}+2x^{3}v(x,a)s^{2}+x^{6}s^{2})^{2}}}-\frac{8x^{7}s\left(1-s^{2}\right)}{2\left(1-x^{8}\right)}\right)\cdot\frac{1}{M\left(x,a\right)\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}
  22. f\left(p,a,b\right)=\frac{p\left(b^{2}+2a^{3}b+a^{6}p^{2}\right)}{b^{2}+2a^{3}bp^{2}+a^{6}p^{2}}
  23. g\left(x,a,b\right)=\frac{d}{dx}\left(f\left(x,a,b\right)\right)
  24. G\left(u,a\right)=g\left(s,u,v\left(u,a\right)\right)
  25. \frac{1}{\sqrt{\left(1-r\left(x,a\right)^{2}\right)\left(1-V\left(x,a\right)r\left(x,a\right)^{2}\right)}}
  26. \frac{1}{G\left(x,a\right)M\left(x,a\right)\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}
  27. E\left(r\left(x,a\right),V\left(x,a\right)\right)-n_{1}M\left(x,a\right)E\left(s,U\left(x,a\right)\right)-\frac{2V\left(x,a\right)\left(1-V\left(x,a\right)\right)}{\frac{d}{dx}\left(V\left(x,a\right)\right)}\left(\left(\frac{d}{dx}\left(\frac{1}{M\left(x,a\right)}\right)\right)-\frac{\left(\frac{d}{dx}\left(U\left(x,a\right)\right)\right)}{2U\left(x,a\right)M\left(x,a\right)}+\frac{\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)}{2V\left(x,a\right)M\left(x,a\right)}\right)F\left(s,U\left(x,a\right)\right)
  28. \frac{n_{1}x\left(1-U\left(x,a\right)\right)M\left(x,a\right)}{4\sqrt{\left(1-s^{2}\right)\left(1-U\left(x,a\right)s^{2}\right)}}\left(\frac{\frac{r\left(x,a\right)\left(1-r\left(x,a\right)^{2}\right)}{2\left(1-V\left(x,a\right)\right)}\left(\frac{d}{dx}\left(V\left(x,a\right)\right)\right)-\left(\frac{d}{dx}\left(r\left(x,a\right)\right)\right)}{G\left(x,a\right)}-\frac{\left(\frac{d}{dx}\left(U\left(x,a\right)\right)\right)s\left(1-s^{2}\right)}{2\left(1-U\left(x,a\right)\right)}\right)
  29. n_{1}=\frac{9}{3}
复制代码



\begin{align*}
&\qquad\qquad\qquad\quad\>\rho\left(x\right)=\frac{x(2\sqrt{3}+(2-\sqrt{3})x^{2})}{2+\sqrt{3}x^{2}}\\
\\
&\qquad\>\sqrt{3}\int_{0}^{x}\sqrt{\frac{1-({\scriptsize\frac{\sqrt{6}-\sqrt{2}}{4}})^{2}t^{2}}{1-t^{2}}}\mathrm{d}t-\int_{0}^{\rho\left(x\right)}\sqrt{\frac{1-({\scriptsize\frac{\sqrt{6}-\sqrt{2}}{4}})^{2}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\scriptsize\frac{\sqrt{6}-\sqrt{2}}{4}})^{2}t^{2})}}-\frac{2x\sqrt{(1-x^{2})(1-({\scriptsize\frac{\sqrt{6}-\sqrt{2}}{4}})^{2}x^{2})}}{2+\sqrt{3}x^{2}}\\
\end{align*}









  1. u=\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^{1/4}
  2. v=\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^{1/4}
  3. \beta\left(x,a\right)=\frac{x\left(2\sqrt{3}+\left(2-\sqrt{3}\right)x^{2}\right)}{2+\sqrt{3}x^{2}}
  4. \frac{4\left(2u^{3}+v\right)^{2}}{3uv^{2}\left(1-u^{8}\right)}\int_{0}^{\beta\left(x,a\right)}\sqrt{\frac{1-v^{8}t^{2}}{1-t^{2}}}dt-\frac{4\left(2u^{3}+v\right)}{uv\left(1-u^{8}\right)}\int_{0}^{x}\sqrt{\frac{1-u^{8}t^{2}}{1-t^{2}}}dt-\left(\frac{2\left(1-v^{8}\right)\left(3u^{3}+2v\right)\left(2u^{3}+v\right)^{2}}{3uv^{3}\left(1-u^{8}\right)}-\frac{2\left(u^{3}+2v\right)}{uv}\right)\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-u^{8}t^{2}\right)}}dt
  5. \frac{6u^{2}\left(2u^{3}+v\right)^{3}x}{(1-u^{8})\sqrt{\frac{1-u^{8}x^{2}}{1-x^{2}}}}\left(\frac{\frac{2v^{5}\left(v^{2}-u^{6}x^{2}\right)\left(v^{2}+2u^{3}v+u^{6}x^{2}\right)}{9u^{3}\left(v^{2}+2u^{3}vx^{2}+u^{6}x^{2}\right)}+\frac{1-v^{8}}{9v^{2}}-\frac{1-u^{8}}{\left(2u^{3}+v\right)^{2}}}{v^{2}+2u^{3}v-2u^{3}vx^{2}-u^{6}x^{2}}-\frac{2u^{5}}{3v\left(2u^{3}+v\right)^{2}}\right)
  6. \sqrt{3}\int_{0}^{x}\sqrt{\frac{1-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^{2}t^{2}}{1-t^{2}}}dt-\int_{0}^{\beta\left(x,a\right)}\sqrt{\frac{1-\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^{2}t^{2}}{1-t^{2}}}dt
  7. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^{2}t^{2}\right)}}dt-\frac{2x\sqrt{\left(1-x^{2}\right)\left(1-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^{2}x^{2}\right)}}{2+\sqrt{3}x^{2}}
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 楼主| 青青子衿 发表于 2024-2-12 22:38
本帖最后由 青青子衿 于 2025-1-19 15:43 编辑
\begin{gather*}
{T\kern-.4ex\lower.48ex\hbox{R}\kern-.4ex}
{T\kern-.4ex\raise.48ex\hbox{R}\kern-.4ex}\\
\\
\frac{\sqrt{5}x(1+\frac{\sqrt{2+2\sqrt{5}}-2}{2}x^{2}+\frac{15-\sqrt{5}-5\sqrt{2+2\sqrt{5}}}{20}x^{4})}{1+\frac{\sqrt{10\sqrt{5}-10}}{2}x^{2}+\frac{3\sqrt{5}-5-\sqrt{50\sqrt{5}-110}}{4}x^{4}}
\end{gather*}

  1. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-(\frac{\sqrt{2}-\sqrt{10}+2\sqrt{\sqrt{5}-1}}{4})^{2}t^{2})}}dt
  2. \frac{1}{\sqrt{5}}\int_{0}^{\frac{\sqrt{5}x(1+\frac{\sqrt{2+2\sqrt{5}}-2}{2}x^{2}+\frac{15-\sqrt{5}-5\sqrt{2+2\sqrt{5}}}{20}x^{4})}{1+\frac{\sqrt{10\sqrt{5}-10}}{2}x^{2}+\frac{3\sqrt{5}-5-\sqrt{50\sqrt{5}-110}}{4}x^{4}}}\frac{1}{\sqrt{(1-t^{2})(1-(\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4})^{2}t^{2})}}dt
复制代码


\begin{gather*}
\rho\left(x\right)=\frac{x(\sqrt{5}+\frac{\sqrt{10+10\sqrt{5}}-2\sqrt{5}}{2}x^{2}+\frac{3\sqrt{5}-1-\sqrt{10+10\sqrt{5}}}{4}x^{4})}{1+\frac{\sqrt{10\sqrt{5}-10}}{2}x^{2}+\frac{3\sqrt{5}-5-\sqrt{50\sqrt{5}-110}}{4}x^{4}}\>\>\>\\
\\
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2})}}=\frac{1}{\sqrt{5}}\int_{0}^{\rho\left(x\right)}
\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2})}}
\\
\\
\begin{split}
&\qquad\sqrt{\small2\sqrt{5}-2}\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2})}}\\
&\qquad\qquad\qquad-\sqrt{5}\int_{0}^{x}\sqrt{\frac{1-({\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\qquad\qquad\qquad\quad+\int_{0}^{\rho\left(x\right)}\sqrt{\frac{1-({\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\frac{\sqrt{\small2\sqrt{5}-2}\>x\left(1+{\small\frac{\sqrt{5}-3+2\sqrt{\sqrt{5}-2}}{2}}x^{2}\right)\sqrt{(1-x^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}x^{2})}}
{1+{\small\frac{\sqrt{10\sqrt{5}-10}}{2}}x^{2}+{\small\frac{3\sqrt{5}-5-\sqrt{50\sqrt{5}-110}}{4}}x^{4}}
\end{split}
\end{gather*}

\begin{gather*}

\rho_5\left(x,s\right)=\frac{x\left(
{\kern-.1ex\raise.3ex\hbox{$\scriptsize1+2s$}\kern-.1ex}
+
\left({\kern-.1ex\raise.3ex\hbox{$\tiny(1+s)\sqrt{(1+2s)(1+s^{2})}-1-2s$}\kern-.1ex}\right)
x^{2}
+
{\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{1+2s+s^{2}+s^{3}-(1+s)\sqrt{(1+2s)(1+s^{2})}}{2}$}\kern-.1ex}
x^{4}\right)}{
1+
\left({\kern-.1ex\raise.3ex\hbox{$\tiny\sqrt{(1+2s)(1+s^{2})}-1+s+s^{2}$}\kern-.1ex}
\right)x^{2}
+
{\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{1-s^{2}+s^{3}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2}$}\kern-.1ex}
x^{4}}\>\>\>\\
\\
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}$}\kern-.1ex}t^{2})}}=\frac{1}{1+2s}\int_{0}^{\rho_5\left(x,s\right)}
\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{(1+2s)^{3}-\left(1-11s-s^{2}\right)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)^{3}}$}\kern-.1ex}t^{2})}}
\\
\\
\begin{split}
&\qquad\qquad\>\int_{0}^{x}\frac{{\scriptsize\frac{2 \left(1-s-s^2\right)+2 \sqrt{(1+2s) (1+s^2)}}{1+2s}}}{\sqrt{(1-t^{2})(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}$}\kern-.1ex}t^{2})}}\mathrm{d}t\\
&\qquad\qquad\qquad-\frac{5}{1+2s}\int_{0}^{x}\sqrt{\frac{1-{\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}$}\kern-.1ex}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\qquad\qquad\qquad+\int_{0}^{\rho_5\left(x,s\right)}\sqrt{\frac{1-{\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{(1+2s)^{3}-\left(1-11s-s^{2}\right)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)^{3}}$}\kern-.1ex}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\frac{{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{2\sqrt{(1+2s)(1+s^{2})}-2(1-s-s^{2})}{1+2s}$}\kern-.1ex} x\left(1+{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{(1+2s)(1+s^{2})}-1-s+s^3}{2+2 s+s^2}$}\kern-.1ex}x^{2}\right)\sqrt{\left(1-x^{2}\right)\left(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}$}\kern-.1ex}x^{2}\right)}}
{1+
\left({\kern-.1ex\raise.3ex\hbox{$\tiny\sqrt{(1+2s)(1+s^{2})}-1+s+s^{2}$}\kern-.1ex}
\right)x^{2}
+
{\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{1-s^{2}+s^{3}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2}$}\kern-.1ex}
x^{4}}
\end{split}
\end{gather*}



  2. s=0.43
  3. \rho(x)=\frac{x\left(1+2s+\left((1+s)\sqrt{(1+2s)(1+s^{2})}-1-2s\right)x^{2}+\frac{1+2s+s^{2}+s^{3}-(1+s)\sqrt{(1+2s)(1+s^{2})}}{2}x^{4}\right)}{1+\left(\sqrt{(1+2s)(1+s^{2})}-1+s+s^{2}\right)x^{2}+\frac{1-s^{2}+s^{3}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2}x^{4}}
  4. \int_{0}^{\rho(x)}\sqrt{\frac{1-\frac{(1+2s)^{3}-(1-11s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)^{3}}t^{2}}{1-t^{2}}}dt-\frac{5}{1+2s}\int_{0}^{x}\sqrt{\frac{1-\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}t^{2}}{1-t^{2}}}dt+\int_{0}^{x}\frac{\frac{2(1-s-s^{2})+2\sqrt{(1+2s)(1+s^{2})}}{1+2s}}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}t^{2}\right)}}dt
  5. \frac{\frac{2\sqrt{(1+2s)(1+s^{2})}-2(1-s-s^{2})}{1+2s}x\left(1+\frac{\sqrt{(1+2s)(1+s^{2})}-1-s+s^{3}}{2+2s+s^{2}}x^{2}\right)\sqrt{\left(1-x^{2}\right)\left(1-\frac{1+2s-(1+s-s^{2})\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}x^{2}\right)}}{1+\left(\sqrt{(1+2s)(1+s^{2})}-1+s+s^{2}\right)x^{2}+\frac{1-s^{2}+s^{3}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2}x^{4}}
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 楼主| 青青子衿 发表于 2024-3-3 01:41
本帖最后由 青青子衿 于 2024-4-5 15:54 编辑
青青子衿 发表于 2024-2-12 22:38
\begin{gather*}
\rho\left(x\right)=\frac{x(\sqrt{5}+\frac{\sqrt{10+10\sqrt{5}}-2\sqrt{5}}{2}x^{2}+\frac{3\sqrt{5}-1-\sqrt{10+10\sqrt{5}}}{4}x^{4})}{1+\frac{\sqrt{10\sqrt{5}-10}}{2}x^{2}+\frac{3\sqrt{5}-5-\sqrt{50\sqrt{5}-110}}{4}x^{4}}\>\>\>\\
\\
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2})}}=\frac{1}{\sqrt{5}}\int_{0}^{\rho\left(x\right)}
\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2})}}
\\
\\
\begin{split}
&\qquad\sqrt{\small2\sqrt{5}-2}\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2})}}\\
&\qquad\qquad\qquad-\sqrt{5}\int_{0}^{x}\sqrt{\frac{1-({\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\qquad\qquad\qquad\quad+\int_{0}^{\rho\left(x\right)}\sqrt{\frac{1-({\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\frac{\sqrt{\small2\sqrt{5}-2}\>x\left(1+{\small\frac{\sqrt{5}-3+2\sqrt{\sqrt{5}-2}}{2}}x^{2}\right)\sqrt{(1-x^{2})(1-({\kern-.1ex\raise.3ex\hbox{$\tiny\frac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4}$}\kern-.1ex})^{2}x^{2})}}
{1+{\small\frac{\sqrt{10\sqrt{5}-10}}{2}}x^{2}+{\small\frac{3\sqrt{5}-5-\sqrt{50\sqrt{5}-110}}{4}}x^{4}}
\end{split}
\end{gather*}



\begin{gather*}
\begin{split}
\Omega_5(x)&=-\int_{0}^{x}\frac{
p_{5}
}{\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\qquad\>+\int_{0}^{x}\frac{
5\varUpsilon_5
}{(1-U_5\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\qquad\>\>\>\>-\int_{0}^{y_5(x)}\frac{M_5q_{5}}{(1-V_5{\raise.5ex\hbox{$\gamma$}}_5^{2}t^{2})\sqrt{(1-t^{2})(1-V_5t^{2})}}\mathrm{d}t\\
&=\frac{\varUpsilon_5\beta}{\sqrt{(1-\beta^{2})(1-U_5\beta^{2})}}\operatorname{artanh}\left(\varrho_5(x)R_5(x)\right)\\
\end{split}\\
\\
\>\>\>\begin{split}
y_{5}(x)&=\tfrac{\frac{1}{M_5}x(1+\tau_{5,1}x^2+\tau_{5,2}x^4)\>}{1+\omega_{5,1}x^2+\omega_{5,2}x^4}\\

\rho_5(x)&=\tfrac{h_5x(1+\sigma_{5,1}x^2)}
{1+
\omega_{5,1}x^{2}+\omega_{5,2}x^{4}
}\\
\varrho_5(x)&=\tfrac{w_3x(1+\varsigma_{5,1}x^2+\varsigma_{5,2}x^4)}{1+\varpi_{5,1}x^2+\varpi_{5,2}x^4+\varpi_{5,3}x^6+\varpi_{5,4}x^8}\\
R_5(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_5\beta^{2})(1-x^{2})(1-U_5x^{2})}}\\
\end{split}\>\\
\end{gather*}



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 楼主| 青青子衿 发表于 2024-3-31 20:36
本帖最后由 青青子衿 于 2024-4-5 16:12 编辑
青青子衿 发表于 2024-3-3 01:41
\begin{gather*}
\begin{split}
\Omega_5(x)&=-\int_{0}^{x}\frac{
p_{5}
}{\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\qquad\>+\int_{0}^{x}\frac{
5\varUpsilon_5
}{(1-U_5\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\qquad\>\>\>\>-\int_{0}^{y_5(x)}\frac{M_5q_{5}}{(1-V_5{\raise.5ex\hbox{$\gamma$}}_5^{2}t^{2})\sqrt{(1-t^{2})(1-V_5t^{2})}}\mathrm{d}t\\
&=\frac{\varUpsilon_5\beta}{\sqrt{(1-\beta^{2})(1-U_5\beta^{2})}}\operatorname{artanh}\left(\varrho_5(x)R_5(x)\right)\\
\end{split}\\
\\
\>\>\>\begin{split}
y_{5}(x)&=\tfrac{\frac{1}{M_5}x(1+\tau_{5,1}x^2+\tau_{5,2}x^4)\>}{1+\omega_{5,1}x^2+\omega_{5,2}x^4}\\

\rho_5(x)&=\tfrac{h_5x(1+\sigma_{5,1}x^2)}
{1+
\omega_{5,1}x^{2}+\omega_{5,2}x^{4}
}\\
\varrho_5(x)&=\tfrac{w_3x(1+\varsigma_{5,1}x^2+\varsigma_{5,2}x^4)}{1+\varpi_{5,1}x^2+\varpi_{5,2}x^4+\varpi_{5,3}x^6+\varpi_{5,4}x^8}\\
R_5(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_5\beta^{2})(1-x^{2})(1-U_5x^{2})}}\\
\end{split}\>\\
\end{gather*}




\begin{gather*}
\\

\begin{split}
I_{2}(x)&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-U_2t^{2}\right)}}\\
&=M_2\int_{0}^{y_{2}(x)}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-V_2t^{2}\right)}}
\end{split}\quad\>\\
\\
\begin{split}
\Xi_2(x)&=\int_{0}^{x}\frac{g_2}{\sqrt{(1-t^{2})(1-U_2t^{2})}}\mathrm{d}t\\
&\quad\>\>\>\>-2M_2\int_{0}^{x}\sqrt{\frac{1-U_2t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\>\>\>\>+\int_{0}^{y_2(x)}\sqrt{\frac{1-V_2t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\rho_2(x)\sqrt{\small(1-x^{2})(1-U_2x^{2})}
\end{split}\qquad\>\>\>\>\\
\\
\begin{split}
\Omega_2(x)&=-\int_{0}^{x}\frac{
p_{2}
}{\sqrt{(1-t^{2})(1-U_2t^{2})}}\mathrm{d}t\\
&\qquad\>+\int_{0}^{x}\frac{
2\varUpsilon_2
}{(1-U_2\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_2t^{2})}}\mathrm{d}t\\
&\qquad\>\>\>\>-\int_{0}^{y_2(x)}\frac{M_2q_{2}}{(1-V_2\gamma_2^{2}t^{2})\sqrt{(1-t^{2})(1-V_2t^{2})}}\mathrm{d}t\\
&=\frac{\varUpsilon_2\beta}{\sqrt{(1-\beta^{2})(1-U_2\beta^{2})}}\operatorname{artanh}\left(\varrho_2(x)R_2(x)\right)\\
\end{split}\\
\\
\>\>\>\begin{split}
y_{2}(x)&=\tfrac{\frac{1}{M_2}x\>\>}{1+\omega_{2,1}x^2}\\
\rho_2(x)&=\tfrac{h_2x}
{1+
\omega_{2,1}x^{2}
}\\
\varrho_2(x)&=\tfrac{w_2x}{1+\varpi_{2,1}x^2+\varpi_{2,2}x^4}\\
R_2(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_2\beta^{2})(1-x^{2})(1-U_2x^{2})}}\\
\end{split}\>\\
\\
\\
\quad\begin{split}
M_2&=\tfrac{1}{1+k}\\
U_2&={\small\>\!k^2}\\
V_2&=\tfrac{4k}{(1+k)^{2}}\\
g_2&={\small\,1-k}\\
h_2&={\scriptsize\,g_{2}-2M_{2}+\tfrac{1}{M_{2}}}\\&={\small\tfrac{2k}{1+k}}\\
\end{split}\qquad
\begin{split}
\omega_{2,1}&={\small\>\!k}\\
\gamma_2&=\tfrac{\frac{1}{M_2}\beta\,\varUpsilon_2}{\varGamma_{2}}\\
p_2&={\small\,P_2}\\
q_2&=\tfrac{Q_{2}}{\varGamma_{2}}\\
\end{split}\quad\>\>\>\\
\\
\qquad
\begin{split}
\varUpsilon_2&={\small\>\!1}\\
\varGamma_{2}&={\small\>\!1+\omega_{2,1}\beta^2}\\
P_2&={\small\>\!1}\\
Q_2&={\small1-k\beta^2}\\
\end{split}\>\>\\
\\
\begin{split}
w_2&=\tfrac{2\varUpsilon_2-p_{2}-q_{2}}{\varUpsilon_2\beta^{2}}={\small\tfrac{\tilde\varGamma_2W_2}{\varGamma_2}}=
{\small\tfrac{2k}{1+k\beta^{2}}}\\
{\small\tilde\varGamma_2}&={\scriptsize(\partial^{2}_{\beta}\varGamma_2)_{\beta=0}}\\
{\small\,\!W_2}&={\small1}\\
\varpi_{2,1}&={\small\tfrac{k(1-2\beta^{2}-2k\beta^{2}-2k^{2}\beta^{2}+k^{2}\beta^{4})}{1+k\beta^{2}}}\\
\varpi_{2,2}&={\scriptsize\,k^{3}\beta^{2}}
\end{split}\quad\>
\end{gather*}


  1. k=0.827
  2. M_{2}=\frac{1}{1+k}
  3. U_{2}=k^{2}
  4. V_{2}=\frac{4k}{(1+k)^{2}}
  5. \omega_{21}=k
  6. y_{2}=\frac{\frac{1}{M_{2}}x}{1+\omega_{21}x^{2}}
  7. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-U_{2}t^{2})}}dt
  8. M_{2}\int_{0}^{y_{2}}\frac{1}{\sqrt{(1-t^{2})(1-V_{2}t^{2})}}dt
  9. g_{2}=1-k
  10. h_{2}=\frac{2k}{1+k}
  11. g_{2}-2M_{2}+\frac{1}{M_{2}}-h_{2}
  12. \int_{0}^{x}\frac{g_{2}}{\sqrt{(1-t^{2})(1-U_{2}t^{2})}}dt-2M_{2}\int_{0}^{x}\sqrt{\frac{1-U_{2}t^{2}}{1-t^{2}}}dt+\int_{0}^{y_{2}}\sqrt{\frac{1-V_{2}t^{2}}{1-t^{2}}}dt
  13. \frac{h_{2}x\sqrt{(1-x^{2})(1-U_{2}x^{2})}}{1+\omega_{21}x^{2}}
  14. \beta=0.509
  15. \gamma_{2}=\frac{\frac{1}{M_{2}}\beta\Upsilon_{2}}{\Gamma_{2}}
  16. \gamma_{2}-\frac{\frac{1}{M_{2}}\beta}{1+k\beta^{2}}
  17. \Upsilon_{2}=1+0
  18. \Gamma_{2}=1+k\beta^{2}
  19. p_{2}=1+0
  20. q_{2}=\frac{Q_{2}}{\Gamma_{2}}
  21. Q_{2}=1-k\beta^{2}
  22. w_{2}=\frac{2k}{1+k\beta^{2}}
  23. \frac{2\Upsilon_{2}-p_{2}-q_{2}}{\Upsilon_{2}\beta^{2}}-w_{2}
  24. \varpi_{21}=\frac{k(1-2\beta^{2}-2k\beta^{2}-2k^{2}\beta^{2}+k^{2}\beta^{4})}{1+k\beta^{2}}
  25. \varpi_{22}=k^{3}\beta^{2}
  26. \int_{0}^{x}\left(\frac{2\Upsilon_{2}}{(1-k^{2}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}-\frac{p_{2}}{\sqrt{(1-t^{2})(1-U_{2}t^{2})}}\right)dt-\int_{0}^{y_{2}}\frac{M_{2}q_{2}}{\left(1-V_{2}\gamma_{2}^{2}\ t^{2}\right)\sqrt{(1-t^{2})(1-V_{2}t^{2})}}dt
  27. \frac{\Upsilon_{2}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{w_{2}x}{1+\varpi_{21}x^{2}+\varpi_{22}x^{4}}\beta\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})(1-x^{2})(1-k^{2}x^{2})}\right)
复制代码




\begin{gather*}
\\

\begin{split}
I_{3}(x)&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-U_3t^{2}\right)}}\\
&=M_3\int_{0}^{y_{3}(x)}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-V_3t^{2}\right)}}
\end{split}\quad\>\\
\\
\begin{split}
\Xi_3(x)&=\int_{0}^{x}\frac{g_3}{\sqrt{(1-t^{2})(1-U_3t^{2})}}\mathrm{d}t\\
&\quad\>\>\>\>-3M_3\int_{0}^{x}\sqrt{\frac{1-U_3t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\>\>\>\>+\int_{0}^{y_3(x)}\sqrt{\frac{1-V_3t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\rho_3(x)\sqrt{\small(1-x^{2})(1-U_3x^{2})}
\end{split}\qquad\>\>\>\>\\
\\
\begin{split}
\Omega_3(x)&=-\int_{0}^{x}\frac{
p_{3}
}{\sqrt{(1-t^{2})(1-U_3t^{2})}}\mathrm{d}t\\
&\qquad\>+\int_{0}^{x}\frac{
3\varUpsilon_3
}{(1-U_3\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_3t^{2})}}\mathrm{d}t\\
&\qquad\>\>\>\>-\int_{0}^{y_3(x)}\frac{M_3q_{3}}{(1-V_3{\raise.3ex\hbox{$\gamma$}}_3^{2}t^{2})\sqrt{(1-t^{2})(1-V_3t^{2})}}\mathrm{d}t\\
&=\frac{\varUpsilon_3\beta}{\sqrt{(1-\beta^{2})(1-U_3\beta^{2})}}\operatorname{artanh}\left(\varrho_3(x)R_3(x)\right)\\
\end{split}\\
\\
\>\>\>\begin{split}
y_{3}(x)&=\tfrac{\frac{1}{M_3}x(1+\tau_{3,1}x^2)\>}{1+\omega_{3,1}x^2}\\
\rho_3(x)&=\tfrac{h_2x}
{1+
\omega_{3,1}x^{2}
}\\
\varrho_3(x)&=\tfrac{w_3x}{1+\varpi_{3,1}x^2+\varpi_{3,2}x^4}\\
R_3(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_3\beta^{2})(1-x^{2})(1-U_3x^{2})}}\\
\end{split}\>\\
\\
\\
\quad\begin{split}
M_3&=\tfrac{1}{1+2s}\\
U_3&={\scriptsize\tfrac{s^{3}\>\!(2+s)}{1+2s}}\\
V_3&={\scriptsize{\raise.4ex\hbox{$\,s\!\left(\tfrac{2+s}{1+2s}\right)\,\!^{3}$}}}\\
g_3&={\small\tfrac{2(1-s^{2})}{1+2s}}\\
h_3&={\scriptsize\,g_{3}-3M_{3}+\tfrac{1}{M_{3}}}\\&={\small\tfrac{2s\>\!(2+s)}{1+2s}}\\

\end{split}\qquad
\begin{split}
\tau_{3,1}&={\small\tfrac{s^2}{1+2s}}\\
\omega_{3,1}&={\scriptsize\,s(2+s)}\\
\gamma_3&=\tfrac{\frac{1}{M_2}\beta\,\varUpsilon_3}{\varGamma_{3}}\\
p_3&={\small\>\!2P_{3}}\\
q_3&=\tfrac{Q_{3}}{\varGamma_{3}}\\

\end{split}\quad\>\>\>\\
\\
\begin{split}\\
\\
\end{split}\qquad\>
\begin{split}
\varUpsilon_3&={\scriptsize1+\tfrac{s^2}{1+2s}\beta^2}\\
\varGamma_{3}&={\scriptsize\>\!1+s(2+s)\beta^2}\\
P_3&={\small\>\!1}\\
Q_3&={\scriptsize1-\tfrac{2s\ (1+s+s^{2})}{1+2s}\beta^{2}+\tfrac{s^{3}(2+s)}{1+2s}\beta^{4}}\\
\end{split}\>\>\\
\\
\qquad\>\>\>\begin{split}
w_3&={\small\tfrac{3\varUpsilon_3-p_{3}-q_{3}}{\varUpsilon_3\beta^{2}}={\small\tfrac{\tilde\varGamma_3W_3}{\varGamma_3}}=
\tfrac{2s(2+s)}{1+s(2+s)\beta^{2}}}\\
{\small\tilde\varGamma_3}&={\scriptsize(\partial^{2}_{\beta}\varGamma_3)_{\beta=0}}\\
{\small\,\!W_3}&={\small1}\\
\varpi_{3,1}&={\small\tfrac{s(2+s)\left(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4}\right)}{(1+2s)\left(1+s(2+s)\beta^{2}\right)}}\\
\varpi_{3,2}&={\small\tfrac{s^{4}(2+s)^{2}\beta^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}\left(1+s(2+s)\beta^{2}\right)}}
\end{split}
\end{gather*}


  1. s=0.402
  2. M_{3}=\frac{1}{1+2s}
  3. U_{3}=\frac{s^{3}(2+s)}{1+2s}
  4. V_{3}=s\frac{(2+s)^{3}}{(1+2s)^{3}}
  5. \tau_{31}=\frac{s^{2}}{1+2s}
  6. \omega_{31}=s(2+s)
  7. y_{3}=\frac{\frac{1}{M_{3}}x\left(1+\tau_{31}x^{2}\right)}{1+\omega_{31}x^{2}}
  8. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-U_{3}t^{2})}}dt
  9. M_{3}\int_{0}^{y_{3}}\frac{1}{\sqrt{(1-t^{2})(1-V_{3}t^{2})}}dt
  10. g_{3}=\frac{2(1-s^{2})}{1+2s}
  11. h_{3}=\frac{2s(2+s)}{1+2s}
  12. g_{3}-3M_{3}+\frac{1}{M_{3}}-h_{3}
  13. \int_{0}^{x}\frac{g_{3}}{\sqrt{(1-t^{2})(1-U_{3}t^{2})}}dt-3M_{3}\int_{0}^{x}\sqrt{\frac{1-U_{3}t^{2}}{1-t^{2}}}dt+\int_{0}^{y_{3}}\sqrt{\frac{1-V_{3}t^{2}}{1-t^{2}}}dt
  14. \frac{h_{3}x\sqrt{(1-x^{2})(1-U_{3}x^{2})}}{1+\omega_{31}x^{2}}
  15. \beta=0.531
  16. \gamma_{3}=\frac{\frac{1}{M_{3}}\beta\Upsilon_{3}}{\Gamma_{3}}
  17. \Upsilon_{3}=1+\frac{s^{2}}{1+2s}\beta^{2}
  18. \Gamma_{3}=1+s(2+s)\beta^{2}
  19. p_{3}=2+0
  20. q_{3}=\frac{Q_{3}}{\Gamma_{3}}
  21. Q_{3}=1-\frac{2s(1+s+s^{2})}{1+2s}\beta^{2}+\frac{s^{3}(2+s)}{1+2s}\beta^{4}
  22. w_{3}=\frac{2s(2+s)}{1+s(2+s)\beta^{2}}
  23. \frac{3\Upsilon_{3}-p_{3}-q_{3}}{\Upsilon_{3}\beta^{2}}-w_{3}
  24. \varpi_{31}=\frac{s(2+s)\left(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4}\right)}{(1+2s)\left(1+s(2+s)\beta^{2}\right)}
  25. \varpi_{32}=\frac{s^{4}(2+s)^{2}\beta^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}\left(1+s(2+s)\beta^{2}\right)}
  26. \int_{0}^{x}\left(\frac{3\Upsilon_{3}}{\left(1-U_{3}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}-\frac{p_{3}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}\right)dt-\int_{0}^{y_{3}}\frac{M_{3}q_{3}}{\left(1-V_{3}\gamma_{3}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}dt
  27. \frac{\Upsilon_{3}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{w_{3}\ x\ }{1+\varpi_{31}x^{2}+\varpi_{32}x^{4}}\beta\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-U_{3}x^{2}\right)}\right)
复制代码




\begin{gather*}
\\

\begin{split}
I_{5}(x)&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-U_5t^{2}\right)}}\\
&=M_5\int_{0}^{y_{5}(x)}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-V_5t^{2}\right)}}
\end{split}\quad\>\\
\\
\begin{split}
\Xi_5(x)&=\int_{0}^{x}\frac{g_5}{\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\quad\>\>\>\>-5M_5\int_{0}^{x}\sqrt{\frac{1-U_5t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\>\>\>\>+\int_{0}^{y_5(x)}\sqrt{\frac{1-V_5t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\rho_5(x)\sqrt{\small(1-x^{2})(1-U_5x^{2})}
\end{split}\qquad\>\>\>\>\\
\\
\begin{split}
\Omega_5(x)&=-\int_{0}^{x}\frac{
p_{5}
}{\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\qquad\>+\int_{0}^{x}\frac{
5\varUpsilon_5
}{(1-U_5\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
&\qquad\>\>\>\>-\int_{0}^{y_5(x)}\frac{M_5q_{5}}{(1-V_5{\raise.5ex\hbox{$\gamma$}}_5^{2}t^{2})\sqrt{(1-t^{2})(1-V_5t^{2})}}\mathrm{d}t\\
&=\frac{\varUpsilon_5\beta}{\sqrt{(1-\beta^{2})(1-U_5\beta^{2})}}\operatorname{artanh}\left(\varrho_5(x)R_5(x)\right)\\
\end{split}\\
\\
\>\>\>\begin{split}
y_{5}(x)&=\tfrac{\frac{1}{M_5}x(1+\tau_{5,1}x^2+\tau_{5,2}x^4)\>}{1+\omega_{5,1}x^2+\omega_{5,2}x^4}\\

\rho_5(x)&=\tfrac{h_5x(1+\sigma_{5,1}x^2)}
{1+
\omega_{5,1}x^{2}+\omega_{5,2}x^{4}
}\\
\varrho_5(x)&=\tfrac{w_3x(1+\varsigma_{5,1}x^2+\varsigma_{5,2}x^4)}{1+\varpi_{5,1}x^2+\varpi_{5,2}x^4+\varpi_{5,3}x^6+\varpi_{5,4}x^8}\\
R_5(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_5\beta^{2})(1-x^{2})(1-U_5x^{2})}}\\
\end{split}\>\\
\\
\\
\quad\>\begin{split}

\delta_{5}&={\scriptsize\sqrt{\raise{0.1ex}\bbox{\Tiny(1+2s)(1+s^{2})}}}\\
M_5&=\tfrac{1}{1+2s}\\

U_5&={\scriptsize\tfrac{1+2s-(1+s-s^{2})\delta_{5}}{2(1+2s)}}\\
V_5&={\scriptsize\tfrac{(1+2s)^{3}-\left(1-11s-s^{2}\right)\delta_{5}}{2(1+2s)^{3}}}\\
g_5&={\small\tfrac{2(1-s-s^2)+2 \delta_{5}}{1+2s}}\\
h_5&={\scriptsize\,g_{5}-5M_{5}+\tfrac{1}{M_{5}}}\\
&={\small\tfrac{2\delta_{5}-2(1-s-s^{2})}{1+2s}}\\

\end{split}\qquad\>
\begin{split}
\tau_{5,1}&={\small\tfrac{(1+s)\delta_{5}-1-2s}{1+2s}}\\
\tau_{5,2}&={\small\tfrac{1+2s+s^{2}+s^{3}-(1+s)\delta_{5}}{2(1+2s)}}\\
\omega_{5,1}&={\scriptsize\,\delta_{5}-1+s+s^{2}}\\
\omega_{5,2}&={\small\,\tfrac{1-s^{2}+s^{3}-(1-s)\delta_{5}}{2}}\\
\gamma_5&=\tfrac{\frac{1}{M_5}\beta\,\varUpsilon_5}{\varGamma_{5}}\\
p_5&={\small\>\!4P_{5}}\\
q_5&=\tfrac{Q_{5}}{\varGamma_{5}}\\
\end{split}\\
\\
\>\>\,
\begin{split}
\sigma_{5,1}&={\small\tfrac{\delta_{5}-1-s+s^3}{2+2 s+s^2}}\\
\varUpsilon_5&={\scriptsize\>\!1+\tfrac{(1+s)\delta_{5}-1-2s}{1+2s}\beta^{2}+\tfrac{1+2s+s^{2}+s^{3}-(1+s)\delta_{5}}{2(1+2s)}\beta^{4}}\\

\varGamma_{5}&={\scriptsize1+{\Tiny(\delta_{5}-1+s+s^{2})}\beta^{2}+\tfrac{1-s^{2}+s^{3}-(1-s)\delta_{5}}{2}\beta^{4}}\\

P_5&={\scriptsize1-\tfrac{1+2s-(1+s)\delta_{5}}{2(1+2s)}\beta^{2}}\\
Q_5&={\scriptsize1+\tfrac{(2+s)\delta_5-2-5s-3s^{2}-2s^{3}}{1+2s}\beta^{2}}\\
&\quad\>\>\>{\scriptsize+\tfrac{3+6s+4s^{2}+5s^{3}-s^{4}-(3+3s+s^{2}-s^{3})\delta_5}{1+2s}\beta^{4}}\\
&\qquad{\scriptsize+\tfrac{(4+4s-s^{4})\delta_5-4-8s-4s^{2}-4s^{3}+s^{4}+s^{5}}{2(1+2s)}\beta^{6}}\\
&\qquad\>\>{\scriptsize+\tfrac{2+4s-2s^{5}+s^{6}-2(1+s-s^{2})\delta_5}{4(1+2s)}\beta^{8}}\\

\end{split}\\
\\
\begin{split}
w_5&={\small\tfrac{5\varUpsilon_5-p_{5}-q_{5}}{\varUpsilon_5\beta^{2}}}={\small\tfrac{\tilde\varGamma_5W_5}{\varGamma_5}}\\
\tilde\varGamma_5&={\small(\partial^{2}_{\beta}\varGamma_5)_{\beta=0}}\\
W_5&={\small\,\!W_{5,1}}\\
W_{5,1}&={\scriptsize1+\tfrac{\delta_{5}-1-s+s^{3}}{2+2s+s^{2}}\beta^{2}}\\
\end{split}
\qquad\>\>\>\begin{split}
\\
\varsigma_{5,1}&={\small\tfrac{S_{5,1}T_{5,1}}{W_5}}\\
\varsigma_{5,2}&={\small\tfrac{S_{5,2}T_{5,2}}{W_5}}\\
\varpi_{5,1}&={\small\tfrac{X_{5,1}Y_{5,1}}{\varGamma_5}}\\
\varpi_{5,2}&={\small\tfrac{X_{5,2}Y_{5,2}}{\varGamma_5}}\\
\varpi_{5,3}&={\small\tfrac{X_{5,3}Y_{5,3}}{\varGamma_5}}\\
\varpi_{5,4}&={\small\tfrac{X_{5,4}Y_{5,4}}{\varGamma_5}}
\end{split}\qquad\\
\\
\begin{split}
S_{5,1}&={\small\tfrac{\delta_{5}-1-s+s^{3}}{2+2s+s^{2}}}\\
T_{5,1}&={\scriptsize1-\tfrac{2+4s+s^{2}\delta_{5}}{1+2s}\beta^{2}+\tfrac{1+2s-(1+s-s^{2})\delta_{5}}{2(1+2s)}\beta^{4}}\\
S_{5,2}&={\small\tfrac{(2+4s-2s^{3}-s^{4}+s^{5})\delta_{5}-(1+2s)(2+2s-s^{4})}{2(1+2s)(2+2s+s^{2})} }\\
T_{5,2}&={\scriptsize\beta^{2}\left(1+\tfrac{(1+s)\delta_{5}-1-2s}{2(1+2s)}\beta^{2}\right)}\\

X_{5,1}&={\scriptsize\delta_{5}-1+s+s^{2}}\\
Y_{5,1}&={\scriptsize1+\tfrac{(3+6s-s^{2}-s^{3}-s^{4})\delta_{5}-(1+2s)(7+7s+s^{2}-3s^{3})}{(1+2s)(2+2s+s^{2})}\beta^{2}}\\
&\qquad\,{\scriptsize+\tfrac{(1+2s)(7-4s^{2})-(7+7s-11s^{2}+4s^{3})\delta_{5}}{2(1+2s)}\beta^{4}}\\
&\qquad\>\>\>\>{\scriptsize+\tfrac{3(2+4s-2s^{3}-s^{4}+s^{5})\delta_{5}-3(1+2s)(2+2s-s^{4})}{2(1+2s)(2+2s+s^{2})}\beta^{6}}\\

X_{5,2}&={\small\tfrac{1-s^{2}+s^{3}-(1-s)\delta_{5}}{2}}\\
Y_{5,2}&={\scriptsize1+\tfrac{(3+3s-4s^{2})\delta_{5}-11(1+2s)}{1+2s}\beta^{2}}\\
&\qquad\,{\scriptsize+\tfrac{17+34s+s^{2}+s^{3}-2s^{5}+2s^{6}-3(3+3s-4s^{2})\delta_{5}}{1+2s}\beta^{4}}\\
&\qquad\>\>\>\>{\scriptsize+\tfrac{(14+14s-15s^{2})\delta_{5}-14-28s+s^{2}+s^{3}+7s^{5}-4s^{6}}{2(1+2s)}\beta^{6}}\\
&\qquad\quad\>\>{\scriptsize+\tfrac{2+4s-2s^{5}+s^{6}-2(1+s-s^{2})\delta_{5}}{4(1+2s)}\beta^{8}}\\

X_{5,3}&={\scriptsize\tfrac{3(2+4s-2s^{2}-2s^{3}+2s^{4}-3s^{5}+2s^{6})-3(2+2s-4s^{2}+2s^{4}-s^{5})\delta_{5}}{2(1+2s)}}\\
Y_{5,3}&={\scriptsize\beta^{2}\left(1+\tfrac{(3+6s-s^{2}-s^{3}-s^{4})\delta_{5}-(1+2s)(7+7s+s^{2}-3s^{3})}{(1+2s)(2+2s+s^{2})}\beta^{2}\right.}\\
&\qquad\>{\scriptsize+\tfrac{(2s+1)(7-4s^{2})-(7+7s-11s^{2}+4s^{3})\delta_{5}}{2(1+2s)}\beta^{4}}\\
&\qquad\>\>\>\>{\scriptsize\left.+\tfrac{3(2+4s-2s^{3}-s^{4}+s^{5})\delta_{5}-3(1+2s)(2+2s-s^{4})}{2(1+2s)(2+2s+s^{2})}\beta^{6}\right)}\\

X_{5,4}&={\scriptsize\tfrac{\left(2+2s-2s^{2}+s^{3}-s^{4}-(2-2s^{2}+s^{3})\delta_{5}\right)^{2}}{16(1+2s)}}\\
Y_{5,4}&={\scriptsize\beta^{4}\left(1+\tfrac{(1+s)\delta_{5}-1-2s}{1+2s}\beta^{2}\right.}\\
&\qquad\quad\>\>\>{\scriptsize\left.+\tfrac{1+2s+s^{2}+s^{3}-(1+s)\delta_{5}}{2(1+2s)}\beta^{4}\right)}\\

\end{split}

\end{gather*}


  1. s=0.597
  2. \delta_{5}=\sqrt{(1+2s)(1+s^{2})}
  3. M_{5}=\frac{1}{1+2s}
  4. U_{5}=\frac{1+2s-(1+s-s^{2})\ \delta_{5}}{2(1+2s)}
  5. V_{5}=\frac{(1+2s)^{3}-(1-11s-s^{2})\ \delta_{5}}{2(1+2s)^{3}}
  6. \tau_{51}=\frac{(1+s)\delta_{5}-1-2s}{1+2s}
  7. \tau_{52}=\frac{1+2s+s^{2}+s^{3}-(1+s)\delta_{5}}{2(1+2s)}
  8. \omega_{51}=\delta_{5}-1+s+s^{2}
  9. \omega_{52}=\frac{1-s^{2}+s^{3}-(1-s)\ \delta_{5}}{2}
  10. y_{5}=\frac{\frac{1}{M_{5}}x\left(1+\tau_{51}x^{2}+\tau_{52}x^{4}\right)}{1+\omega_{51}x^{2}+\omega_{52}x^{4}}
  11. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-U_{5}t^{2})}}dt
  12. M_{5}\int_{0}^{y_{5}}\frac{1}{\sqrt{(1-t^{2})(1-V_{5}t^{2})}}dt
  13. g_{5}=\frac{2(1-s-s^{2})+2\delta_{5}}{1+2s}
  14. h_{5}=\frac{2\delta_{5}-2(1-s-s^{2})}{1+2s}
  15. \sigma_{51}=\frac{\delta_{5}-1-s+s^{3}}{2+2s+s^{2}}
  16. \int_{0}^{x}\frac{g_{5}}{\sqrt{(1-t^{2})(1-U_{5}t^{2})}}dt-5M_{5}\int_{0}^{x}\sqrt{\frac{1-U_{5}t^{2}}{1-t^{2}}}dt+\int_{0}^{y_{5}}\sqrt{\frac{1-V_{5}t^{2}}{1-t^{2}}}dt
  17. \frac{h_{5}x\left(1+\sigma_{51}x^{2}\right)\sqrt{(1-x^{2})(1-U_{5}x^{2})}}{1+\omega_{51}x^{2}+\omega_{52}x^{4}}
  18. \beta=0.949
  19. \delta_{51}=\sqrt{1+2s}
  20. \delta_{52}=\sqrt{1+s^{2}}
  21. \gamma_{5}=\frac{\frac{1}{M_{5}}\beta\ \Upsilon_{5}}{\Gamma_{5}}
  22. \Upsilon_{5}=1+\tau_{51}\beta^{2}+\tau_{52}\beta^{4}
  23. \left(1-\frac{1+2s+s^{2}\delta_{51}-(1+s)\delta_{5}}{2(1+2s)}\beta^{2}\right)\left(1-\frac{1+2s-s^{2}\delta_{51}-(1+s)\delta_{5}}{2(1+2s)}\beta^{2}\right)
  24. \Gamma_{5}=1+\omega_{51}\beta^{2}+\omega_{52}\beta^{4}
  25. \left(1-\frac{1-s-s^{2}+s\delta_{51}+s\delta_{52}-\delta_{5}}{2}\beta^{2}\right)\left(1-\frac{1-s-s^{2}-s\delta_{51}-s\delta_{52}-\delta_{5}}{2}\beta^{2}\right)
  26. p_{5}=4P_{5}
  27. P_{5}=1-\frac{1+2s-(1+s)\ \delta_{5}}{2(1+2s)}\beta^{2}
  28. q_{5}=\frac{Q_{5}}{\Gamma_{5}}
  29. Q_{5}=1+\frac{(2+s)\delta_{5}-2-5s-3s^{2}-2s^{3}}{1+2s}\beta^{2}+\frac{3+6s+4s^{2}+5s^{3}-s^{4}-(3+3s+s^{2}-s^{3})\delta_{5}}{1+2s}\beta^{4}+\frac{(4+4s-s^{4})\delta_{5}-4-8s-4s^{2}-4s^{3}+s^{4}+s^{5}}{2(1+2s)}\beta^{6}+\frac{2+4s-2s^{5}+s^{6}-2(1+s-s^{2})\delta_{5}}{4(1+2s)}\beta^{8}
  30. \left(1-\frac{1+2s+s(2-s)\delta_{51}-(1-s)\ \delta_{5}}{2(1+2s)}\beta^{2}\right)\left(1-\frac{1+2s-s(2-s)\delta_{51}-(1-s)\ \delta_{5}}{2(1+2s)}\beta^{2}\right)\left(1-\frac{1+s+s^{2}+s\delta_{51}-s\ \delta_{52}-\ \delta_{5}}{2}\beta^{2}\right)\left(1-\frac{1+s+s^{2}-s\delta_{51}+s\delta_{52}-\ \delta_{5}}{2}\beta^{2}\right)
  31. \int_{0}^{x}\frac{5\Upsilon_{5}}{(1-U_{5}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_{5}t^{2})}}dt-\int_{0}^{x}\frac{p_{5}}{\sqrt{(1-t^{2})(1-U_{5}t^{2})}}dt-\int_{0}^{y_{5}}\frac{M_{5}q_{5}}{(1-V_{5}\gamma_{5}^{2}t^{2})\sqrt{(1-t^{2})(1-V_{5}t^{2})}}dt
  32. \frac{\Upsilon_{5}\beta}{\sqrt{(1-\beta^{2})(1-U_{5}\beta^{2})}}\operatorname{artanh}\left(\frac{w_{5}x(1+\varsigma_{51}x^{2}+\varsigma_{52}x^{4})}{1+\varpi_{51}x^{2}+\varpi_{52}x^{4}+\varpi_{53}x^{6}+\varpi_{54}x^{8}}\beta\sqrt{(1-\beta^{2})(1-U_{5}\beta^{2})(1-x^{2})(1-U_{5}x^{2})}\right)
  33. w_{5}=\frac{2\left(\delta_{5}-1+s+s^{2}\right)W_{51}}{\Gamma_{5}}
  34. W_{51}=1+\frac{\delta_{5}-1-s+s^{3}}{2+2s+s^{2}}\beta^{2}
  35. \varsigma_{51}=\frac{S_{51}T_{51}}{W_{51}}
  36. \varsigma_{52}=\frac{S_{52}T_{52}}{W_{51}}
  37. \varpi_{51}=\frac{X_{51}Y_{51}}{\Gamma_{5}}
  38. \varpi_{52}=\frac{X_{52}Y_{52}}{\Gamma_{5}}
  39. \varpi_{53}=\frac{X_{53}Y_{53}}{\Gamma_{5}}
  40. \varpi_{54}=\frac{X_{54}Y_{54}}{\Gamma_{5}}
  41. S_{51}=\frac{\delta_{5}-1-s+s^{3}}{2+2s+s^{2}}
  42. T_{51}=1-\frac{2+4s+s^{2}\delta_{5}}{1+2s}\beta^{2}+\frac{1+2s-(1+s-s^{2})\delta_{5}}{2(1+2s)}\beta^{4}
  43. S_{52}=\frac{(2+4s-2s^{3}-s^{4}+s^{5})\delta_{5}-(1+2s)(2+2s-s^{4})}{2(1+2s)(2+2s+s^{2})}
  44. T_{52}=\beta^{2}\left(1+\frac{(1+s)\delta_{5}-1-2s}{2(1+2s)}\beta^{2}\right)
  45. X_{51}=\delta_{5}-1+s+s^{2}
  46. Y_{51}=1+\frac{(3+6s-s^{2}-s^{3}-s^{4})\delta_{5}-(1+2s)(7+7s+s^{2}-3s^{3})}{(1+2s)(2+2s+s^{2})}\beta^{2}+\frac{(1+2s)(7-4s^{2})-(7+7s-11s^{2}+4s^{3})\delta_{5}}{2(1+2s)}\beta^{4}+\frac{3(2+4s-2s^{3}-s^{4}+s^{5})\delta_{5}-3(1+2s)(2+2s-s^{4})}{2(1+2s)(2+2s+s^{2})}\beta^{6}
  47. X_{52}=\frac{1-s^{2}+s^{3}-(1-s)\delta_{5}}{2}
  48. Y_{52}=1+\frac{(3+3s-4s^{2})\delta_{5}-11(1+2s)}{1+2s}\beta^{2}+\frac{17+34s+s^{2}+s^{3}-2s^{5}+2s^{6}-3(3+3s-4s^{2})\delta_{5}}{1+2s}\beta^{4}+\frac{(14+14s-15s^{2})\ \delta_{5}-14-28s+s^{2}+s^{3}+7s^{5}-4s^{6}}{2(1+2s)}\beta^{6}+\frac{2+4s-2s^{5}+s^{6}-2(1+s-s^{2})\delta_{5}}{4(1+2s)}\beta^{8}
  49. X_{53}=\frac{3(2+4s-2s^{2}-2s^{3}+2s^{4}-3s^{5}+2s^{6})-3(2+2s-4s^{2}+2s^{4}-s^{5})\delta_{5}}{2(1+2s)}
  50. Y_{53}=\beta^{2}\left(1+\frac{(3+3s-4s^{2})\delta_{5}-11(1+2s)}{6(1+2s)}\beta^{2}+\frac{5+10s-s^{2}-s^{3}-s^{5}-(5+5s-4s^{2})\delta_{5}}{6(1+2s)}\beta^{4}+\frac{(2+2s-s^{2})\delta_{5}-2-4s-s^{2}-s^{3}+s^{5}}{12(1+2s)}\beta^{6}\right)
  51. X_{54}=\frac{\left(2+2s-2s^{2}+s^{3}-s^{4}-(2-2s^{2}+s^{3})\delta_{5}\right)^{2}}{16(1+2s)}
  52. Y_{54}=\beta^{4}\left(1+\frac{(1+s)\delta_{5}-1-2s}{1+2s}\beta^{2}+\frac{1+2s+s^{2}+s^{3}-(1+s)\delta_{5}}{2(1+2s)}\beta^{4}\right)
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 楼主| 青青子衿 发表于 2024-8-29 12:43
本帖最后由 青青子衿 于 2024-12-21 22:35 编辑
\begin{align*}
\lambda(\frac{1+\sqrt{7}i}{2})&=\frac{2\sqrt{\sqrt{\frac{\lambda(\sqrt{7}i)}{\lambda(\sqrt{7}i)-1}}}}{1+\sqrt{\frac{\lambda(\sqrt{7}i)}{\lambda(\sqrt{7}i)-1}}}\\
\\
\lambda(-\frac{2}{1+\sqrt{7}i})&=\frac{1-\sqrt{\frac{\lambda(\sqrt{7}i)}{\lambda(\sqrt{7}i)-1}}}{1+\sqrt{\frac{\lambda(\sqrt{7}i)}{\lambda(\sqrt{7}i)-1}}}
\end{align*}


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

\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{0.4pt}{\scriptsize\sqrt{2}-1})^2t^{2})}}\\
\\
=\frac{1}{i \sqrt{2}}{\Large{\int}}_{0}^{\small\frac{i\sqrt{2}x}{\sqrt{(1-x^{2})(1-(\raise{0.4pt}{\scriptsize\sqrt{2}-1})^2x^{2})}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.4pt}{\scriptsize\sqrt{2}-1})^2t^{2})}}
\end{gather*}


\begin{align*}

\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}}\\
\\
\int_{0}^{x}\frac{\mathrm{dt}}{\sqrt{1+t^{3}}}&=\dfrac{1}{3^{1/4}}\int_{0}^{\frac{2\cdot3^{1/4}x(\sqrt{1+x}+\sqrt{1-x+x^{2}})}{(1+\sqrt{3})x^{2}+2(1+\sqrt{1+x^{3}})}}\frac{\mathrm{dt}}{\sqrt{(1-t^2)(1-\frac{2+\sqrt{3}}{4}t^2)}}\\
&=\int_{0}^{\frac{2(\sqrt{1+x^{3}}-1)}{x^{2}}}\frac{\mathrm{dt}}{\sqrt{1-t^3}}\\

\end{align*}
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