William Wells Adams双有理变换

青青子衿

\begin{gather*}
\begin{split}
y^2&=x^4+a x^3+b x^2+c x+d\\
v^2&=u^3+A u+B\\
q^2&=p^3+A p+B\\
\end{split}\\
\\
\left\{\begin{split}
x&=\frac{v+q}{u-p}-\frac{a}{4}\\
y&=2u+p-\left(\frac{v+q}{u-p}\right)^2\\
b&=\frac{3}{8} \left(a^2-16 p\right)\\
c&=\frac{1}{16} \left(a^3-48 a p-128 q\right)\\
d&=\frac{1}{256} \left(a^4-96 a^2 p-512 a q-1024 A-768 p^2\right)\\
\end{split}\right.\\
\\
\left\{\begin{split}
u&=\frac{1}{12} \left(3 a x+b+6 x^2+6 y\right)\\
v&=\frac{1}{8} \left(3 a x^2+a y+2 b x+c+4 x^3+4 x y\right)\\
A&=-\frac{1}{48} \left(b^2-3 a c+12 d\right)\\
B&=\frac{1}{1728}(27 a^2 d-9 a b c+2 b^3-72 b d+27 c^2)\\
p&=\frac{1}{48} \left(3 a^2-8 b\right)\\
q&=-\frac{1}{64} \left(a^3-4 a b+8 c\right)\\
\end{split}\right.
\end{gather*}




Multiples of Points on Elliptic Curves and Continued Fractions
researchgate.net/publication/31389206

1. d + c x + b x^2 + a x^3 + x^4 -
2.     y^2 /. {x -> (v - 1/64 (a^3 - 4 a*b + 8 c))/(
3.       u - 1/48 (3 a^2 - 8 b)) - a/4,
4.     y -> 2 u + (3 a^2 - 8 b)/
5.       48 - ((v - 1/64 (a^3 - 4 a*b + 8 c))/(
6.        u - 1/48 (3 a^2 - 8 b)))^2} /. {a -> 12, b -> 942, c -> -17604,
7.     d -> -2151} // Factor
8. u^3 + A*u + B - v^2 /. {u -> (b + 3 a*x + 6 x^2 + 6 y)/12 ,
9.    v -> (c + 2 b*x + 3 a*x^2 + 4 x^3 + a*y + 4 x*y)/8 ,
10.    A -> -((b^2 - 3 a c + 12 d)/48) ,
11.    B -> (2 b^3 - 9 a b c + 27 c^2 + 27 a^2 d - 72 b d)/1728} // Factor

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青青子衿

\begin{gather*}
\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}=\frac{2}{u}\int_{\frac{4(6x^{2}+3\alpha x+\beta+6\sqrt{x^{4}+\alpha x^{3}+\beta x^{2}+\gamma x+\delta})}{3u^{2}}}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{3}+At+B}}\\

\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{3}+At+B}}=\frac{u}{2}\int_{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{x^{3}+Ax+B})}{4(3u^{2}x-w)}}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}\\

\\
\left\{
\begin{split}
A&=\frac{16v}{3u^{4}}=\tfrac{16 \left(3 \alpha  \gamma -\beta ^2-12 \delta \right)}{3 \left(\alpha ^3-4 \alpha  \beta +8 \gamma \right)^4}\\
B&=\frac{1}{u^{4}}\left(1-\frac{48vw}{27u^{2}}-\frac{w^{3}}{27u^{2}}\right)\\
&=
\tfrac{64 \left(27 \alpha ^2 \delta -9 \alpha  \beta  \gamma +2 \beta ^3-72 \beta  \delta +27 \gamma ^2\right)}{27 \left(\alpha ^3-4 \alpha  \beta +8 \gamma \right)^6}\\
u&=\alpha^{3}-4\alpha\beta+8\gamma\\
v&=3\alpha\gamma-\beta^{2}-12\delta\\
w&=3\alpha^{2}-8\beta
\end{split}

\right.
\end{gather*}

青青子衿

青青子衿 发表于 2025-3-8 17:02
\begin{gather*}
\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}=\frac{2}{u}\int_{\frac{4(6x^{2}+3\alpha x+\beta+6\sqrt{x^{4}+\alpha x^{3}+\beta x^{2}+\gamma x+\delta})}{3u^{2}}}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{3}+At+B}}\\

\end{gather*}

\begin{gather*}
\qquad\qquad\int_{x}^{S}\frac{\frac{1}{t-\frac{w}{3u^{2}}}+G}{\sqrt{t^{3}+At+B}}{\mathrm{d}t}\qquad\qquad(S\gg{0})\\
=\int_{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{x^{3}+Ax+B})}{4(3u^{2}x-w)}}^{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{S^{3}+AS+B})}{4(3u^{2}S-w)}}
\frac{\frac{u^{2}}{2}}{t-\frac{(4\beta-w)^{2}-576\delta}{72u}}\left(1-\frac{\frac{24t^{2}+12\alpha t+4\beta-w}{24}-\frac{G}{u}\left(t-\frac{(4\beta-w)^{2}-576\delta}{72u}\right)}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}\right)
{\mathrm{d}t} \\
\\
\qquad\quad\int_{x}^{T}\frac{t+H}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}{\mathrm{d}t}\qquad\qquad(T\gg{0})\\

=\int_{\frac{4(6x^{2}+3\alpha x+\beta+6\sqrt{x^{4}+\alpha x^{3}+\beta x^{2}+\gamma x+\delta})}{3u^{2}}}^{\frac{4(6T^{2}+3\alpha T+\beta+6\sqrt{T^{4}+\alpha T^{3}+\beta T^{2}+\gamma T+\delta})}{3u^{2}}}
\frac{1}{2u^{2}}\left(\frac{u^{2}}{t-\frac{w}{3u^{2}}}-\frac{\frac{1}{t-\frac{w}{3u^{2}}}+(\alpha-4H)u}{\sqrt{t^{3}+At+B}}\right)
{\mathrm{d}t}\\
\\
\left\{
\begin{split}
A&=\frac{16v}{3u^{4}}=\tfrac{16 \left(3 \alpha  \gamma -\beta ^2-12 \delta \right)}{3 \left(\alpha ^3-4 \alpha  \beta +8 \gamma \right)^4}\\
B&=\frac{1}{u^{4}}\left(1-\frac{48vw}{27u^{2}}-\frac{w^{3}}{27u^{2}}\right)\\
&=
\tfrac{64 \left(27 \alpha ^2 \delta -9 \alpha  \beta  \gamma +2 \beta ^3-72 \beta  \delta +27 \gamma ^2\right)}{27 \left(\alpha ^3-4 \alpha  \beta +8 \gamma \right)^6}\\
u&=\alpha^{3}-4\alpha\beta+8\gamma\\
v&=3\alpha\gamma-\beta^{2}-12\delta\\
w&=3\alpha^{2}-8\beta
\end{split}\right.
\end{gather*}

1. \int_{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{x^{3}+Ax+B})}{4(3u^{2}x-w)}}^{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{S^{3}+AS+B})}{4(3u^{2}S-w)}}\frac{\frac{u^{2}}{2}}{t-\frac{(4\beta-w)^{2}-576\delta}{72u}}\left(1-\frac{\frac{24t^{2}+12\alpha t+4\beta-w}{24}-\frac{G}{u}\left(t-\frac{(4\beta-w)^{2}-576\delta}{72u}\right)}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}\right)dt
2. \int_{x}^{S}\frac{\frac{1}{t-\frac{w}{3u^{2}}}+G}{\sqrt{t^{3}+At+B}}dt
3. \int_{x}^{S}\frac{G}{\sqrt{t^{3}+At+B}}dt
4. \int_{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{x^{3}+Ax+B})}{4(3u^{2}x-w)}}^{-\frac{\alpha}{4}-\frac{3u(1-u^{2}\sqrt{S^{3}+AS+B})}{4(3u^{2}S-w)}}\frac{\frac{u}{2}G}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}dt
5. G=1.7
6. S=100
7. A=\frac{16v}{3u^{4}}
8. B=\frac{1}{u^{4}}\left(1-\frac{48vw}{27u^{2}}-\frac{w^{3}}{27u^{2}}\right)
9. u=\alpha^{3}-4\alpha\beta+8\gamma
10. v=3\alpha\gamma-\beta^{2}-12\delta
11. w=3\alpha^{2}-8\beta
12. \alpha=4.9
13. \beta=3.1
14. \gamma=3.4
15. \delta=5.7
16. \int_{\frac{4(6x^{2}+3\alpha x+\beta+6\sqrt{x^{4}+\alpha x^{3}+\beta x^{2}+\gamma x+\delta})}{3u^{2}}}^{\frac{4(6T^{2}+3\alpha T+\beta+6\sqrt{T^{4}+\alpha T^{3}+\beta T^{2}+\gamma T+\delta})}{3u^{2}}}\frac{1}{2u^{2}}\left(\frac{u^{2}}{t-\frac{w}{3u^{2}}}-\frac{\frac{1}{t-\frac{w}{3u^{2}}}+(\alpha-4H)u}{\sqrt{t^{3}+At+B}}\right)dt
17. \int_{x}^{T}\frac{t+H}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}dt
18. \int_{\frac{4(6x^{2}+3\alpha x+\beta+6\sqrt{x^{4}+\alpha x^{3}+\beta x^{2}+\gamma x+\delta})}{3u^{2}}}^{\frac{4(6T^{2}+3\alpha T+\beta+6\sqrt{T^{4}+\alpha T^{3}+\beta T^{2}+\gamma T+\delta})}{3u^{2}}}\frac{\frac{2H}{u}}{\sqrt{t^{3}+At+B}}dt
19. \int_{x}^{T}\frac{H}{\sqrt{t^{4}+\alpha t^{3}+\beta t^{2}+\gamma t+\delta}}dt
20. H=1.6
21. T=100

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\begin{gather*}
\int\frac{\frac{1}{t-\frac{w}{3u^{2}}}+(\frac{16 v+w^2}{6u}-4\ell)u}{\sqrt{t^{3}+At+B}}
{\mathrm{d}t}\\
\\
\left\{
\begin{split}
A&=\tfrac{16v}{3u^{4}}\\
B&=\tfrac{1}{u^{4}}\left(1-\tfrac{48vw}{27u^{2}}-\tfrac{w^{3}}{27u^{2}}\right)\\
P&=\tfrac{16 v+w^2}{12 u}\\
Q&=\tfrac{\left(16 v+w^2\right)^2}{576 u^2}-\tfrac{w}{16}\\
R&=\tfrac{u}{32}\\
\nu_n&=\tfrac{4PR\cdot\,\!\nu_{n-1}-\nu_{n-1}^2\nu_{n-2}+8R^2}{\nu_{n-1}^2}\\ \nu_3&=0\\ \nu_4&=2Q\\
\ell&=\tfrac{P}{n}+\tfrac{1}{4 n R}\sum _{k=3}^n \nu_{k}\nu_{k+1}
\end{split}\right.
\end{gather*}
\begin{align*}
\end{align*}




\begin{gather*}
{\large\int}\frac{\frac{1}{t-\frac{\xi_{1}}{\xi_{2}}}+\bar{\ell}}{\sqrt{t^{3}+\tfrac{3\xi_{3}}{\xi_{2}^{2}}t+\tfrac{9}{\xi_{2}^{2}}\left(1-\tfrac{\xi_{1}\xi_{3}}{3\xi_{2}}-\tfrac{\xi_{1}^{3}}{9\xi_{2}}\right)}}
{\mathrm{d}t}\\
\\
\left\{
\begin{split}
A&=\tfrac{3\xi_{3}}{\xi_{2}^{2}}\\
B&=\tfrac{9}{\xi_{2}^{2}}\left(1-\tfrac{\xi_{1}\xi_{3}}{3\xi_{2}}-\tfrac{\xi_{1}^{3}}{9\xi_{2}}\right)\\

\nu_n&=\tfrac{\tfrac{\xi_{3}+\xi_{1}^2}{96 }\cdot\,\!\nu_{n-1}-\nu_{n-1}^2\nu_{n-2}+\tfrac{\xi_{2}}{384}}{\nu_{n-1}^2}\\ \nu_3&=0\\
\nu_4&=\tfrac{\left( \xi_{3}+\xi_{1}^2\right)^2}{96\xi_{2}}-\tfrac{\xi_{1}}{8}\\

\bar{\ell}&=\tfrac{n-2}{6n}(\xi_{3}+\xi_{1}^2)-\tfrac{32}{n}\sum _{k=3}^n \nu_{k}\nu_{k+1}
\end{split}\right.
\end{gather*}

1. Solve[363 == u/v && -16227 == (3 w)/v^2 &&
2.   7338654 == 9/v^2 (1 - (u*w)/(3*v) - u^3/(9*v)), {u, v, w}]
3. vi = RecurrenceTable[{v[m] == (V + W*v[m - 1] - v[m - 2]*v[m - 1]^2)/
4.        v[m - 1]^2, v[3] == 0, v[4] == U}, v, {m, 3, 12}] // Factor;
5. Thread[v /@ Range[3, Length[vi] + 2] -> vi];
6. (((n - 2) ( 96 W))/(6 n) - (32/n) Sum[v[k]*v[k + 1], {k, 3, n}] /.
7.     n -> 6) /. Thread[v /@ Range[3, Length[vi] + 2] -> vi] // Factor
8. % /. {V -> v/384,
9.     W -> (w + u^2)/96,
10.     U -> -u/8 + ( w + u^2)^2/(96 v)} /. {u -> -(121/780),
11.    v -> -(1/2340), w -> -(601/608400)} // Factor

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