区间变换和余模变换以及虚模变换

青青子衿

椭圆积分的一些区间变换公式
\begin{align*}
\int_{0}^{|x|}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}
&=\int_{\sqrt{\frac{1-x^{2}}{1-k^{2}x^{2}}}}^{1}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\qquad(-1<x<1;0<k<1)\\
&=\int_{1}^{\frac{1}{\sqrt{1-k^{2}x^{2}}}}\frac{\mathrm{d}t}{\sqrt{\left(t^{2}-1\right)\left[1-\left(1-k^{2}\right)t^{2}\right]}}\\
&=\int_{\sqrt{\frac{1-k^{2}x^{2}}{1-k^{2}}}}^{\frac{1}{\sqrt{1-k^{2}}}}\frac{\mathrm{d}t}{\sqrt{\left(t^{2}-1\right)\left[1-\left(1-k^{2}\right)t^{2}\right]}}\\

&=\int_{\frac{1}{k}}^{\frac{1}{k}\sqrt{\frac{1-k^{2}x^{2}}{1-x^{2}}}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\

&=\int_{\frac{1}{k\left|x\right|}}^{+\infty}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
\\


\end{align*}

青青子衿

\begin{align*}
\operatorname{sn}\!\left(\xi,k\mathrm{i}\right)
&=\frac{1}{ \sqrt{1+k^2}}\operatorname{sd}\left(\sqrt{1+k^2} \,\xi,\frac{k}{\sqrt{1+k^2}}\right)\\
\\
\operatorname{sn}^{-1}\!\left(x,k\mathrm{i}\right)&=\frac{1}{ \sqrt{1+k^2}}\operatorname{sn}^{-1}\!\left(x\sqrt{\frac{1+k^2}{1+k^2x^2}},\frac{k}{\sqrt{1+k^2}}\right)\\


\end{align*}

\begin{align*}

\int_0^{x}\frac{1}{\sqrt{(1-t^2) \left(1-(i\kappa  t)^2\right)}}
&=\dfrac{1}{i}\int_0^{ix}\frac{1}{\sqrt{(1+t^2) \left(1+(i\kappa  t)^2\right)}}\\

&=\dfrac{1}{i\sqrt{1+\kappa ^2}}\int_0^{\frac{ix\sqrt{1+\kappa ^2}}{\sqrt{1-x^2}}}\frac{1}{\sqrt{\left(1-t^2\right) \big(1-(\left.t \middle/\sqrt{1+\kappa ^2}\right.\,)^2\big)}}\\
&=\dfrac{1}{\sqrt{1+\kappa ^2}}\int_0^{\frac{x\sqrt{1+\kappa ^2}}{\sqrt{1+\kappa^2x^2}}}\frac{1}{\sqrt{\left(1-t^2\right) \big(1-(\left.\kappa\,t \middle/\sqrt{1+\kappa ^2}\right.\,)^2\big)}}\\

\end{align*}



\begin{align*}
\operatorname{sn}^{-1}\left(x,\sqrt{2}-1\right)&=\dfrac{1}{\sqrt{2}}\operatorname{sn}^{-1}\left(\frac{\sqrt{2}x}{1+\big(\sqrt{2}-1\big)x^{2}},\sqrt{2\big(\sqrt{2}-1\big)}\right)
\\
&=\dfrac{1+\sqrt{2}}{\sqrt{2}}\operatorname{sn}^{-1}\left(\frac{\big(2-\sqrt{2}\big)x}{1-\big(\sqrt{2}-1\big)x^{2}},\sqrt{2\big(1+\sqrt{2}\big)}i\right)
\\
&=\frac{1+\sqrt{2}}{i}\operatorname{sn}^{-1}\left(\frac{i\big(\sqrt{2}-1\big)x}{\sqrt{1-\big(\sqrt{2}-1\big)^{2}x^{2}}},\sqrt{2\big(1+\sqrt{2}\big)}i\right)
\\
&=\frac{1}{\sqrt{2}\,i}\text{sn}^{-1}\left(\frac{\sqrt{2}\,ix}{\sqrt{\big(1-x^2\big) \big(1-\big(\sqrt{2}-1\big)^2 x^2\big)}},\sqrt{2}-1\right)

\end{align*}

1. NumberForm[InverseJacobiSN[x, (Sqrt[2] - 1)^2] /. {x -> 0.1}, 15]
2. NumberForm[
3. 1/Sqrt[2]
4.     InverseJacobiSN[(Sqrt[2] x)/(1 + (Sqrt[2] - 1) x^2),
5.     1 - (Sqrt[2] - 1)^2] /. {x -> 0.1}, 15]
6. NumberForm[(1 + Sqrt[2])/Sqrt[2]
7.     InverseJacobiSN[((2 - Sqrt[2]) x)/(
8.     1 - (Sqrt[2] - 1) x^2), -(Sqrt[2 (1 + Sqrt[2])])^2] /. {x ->
9.     0.1}, 15]
10. NumberForm[(1 + Sqrt[2])/
11.    I InverseJacobiSN[(I (Sqrt[2] - 1) x)/Sqrt[
12.     1 - (Sqrt[2] - 1)^2 x^2], -(Sqrt[2 (1 + Sqrt[2])])^2] /. {x ->
13.     0.1}, 15]
14. NumberForm[
15. 1/(Sqrt[2] I)
16.     InverseJacobiSN[(Sqrt[2] I*x)/
17.     Sqrt[(1 - x^2) (1 - (Sqrt[2] - 1)^2 x^2)], (Sqrt[2] -
18.       1)^2] /. {x -> 0.1}, 15]
19. NIntegrate[1/Sqrt[(1 - (I*X)^2) (1 - ((
20.       Sqrt[10] - Sqrt[2] + 2 Sqrt[Sqrt[5] - 1])/4)^2 (I*X)^2)], {X, 0,
21.    1}]
22. NIntegrate[
23. D[x/Sqrt[1 + x^2],
24.   x]/Sqrt[(1 - (x/Sqrt[
25.      1 + x^2])^2) (1 - ((Sqrt[2] - Sqrt[10] + 2 Sqrt[Sqrt[5] - 1])/
26.       4)^2 (x/Sqrt[1 + x^2])^2)], {x, 0, 1}]

复制代码

青青子衿

\begin{align*}
\Pi\big({\scriptsize\frac{\left(1+k\right)x}{1+kx^{2}}},{\scriptsize{\frac{k+\beta^{2}}{(1+k)\beta}}},{\scriptsize\frac{2\sqrt{k}}{1+k}}\big)&=\int_{0}^{\frac{\left(1+k\right)x}{1+kx^{2}}}\frac{\mathrm{d}t}{\big(1-\big({\scriptsize{\frac{k+\beta^{2}}{(1+k)\beta}}}t\big)^{2}\big)\sqrt{\left(1-t^{2}\right)\big(1-\left({\scriptsize{\frac{2\sqrt{k}}{1+k}}}t\right)^{2}\big)}}\\
&=\int_{0}^{\frac{\left(1+k\right)x}{1+kx^{2}}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-({\scriptsize\frac{2\sqrt{k}}{1+k}}t)^{2}\right)}}\\&\qquad
-\int_{\frac{1+kx^{2}}{2\sqrt{k}x}}^{+\infty}\frac{\mathrm{d}t}{\left(1-({\scriptsize{\frac{2\sqrt{k}\beta}{k+\beta^2}}}t)^2\right)\sqrt{\left(1-t^{2}\right)\left(1-({\scriptsize\frac{2\sqrt{k}}{1+k}}t)^{2}\right)}}
\end{align*}

\begin{align*}
\Pi(x,k\beta,k)&=\int_{0}^{x}\frac{\mathrm{d}t}{\left(1-k^2\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}
\\
&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}-\int_{\frac{1}{kx}}^{+\infty}\frac{\mathrm{d}t}{\left(1-\frac{t^{2}}{\beta^{2}}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}
\end{align*}


\begin{align*}
\Pi(x,k\beta,k)&=\int_{0}^{x}\frac{{\mathrm{d}}t}{(1-k^{2}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}\\
&=\int_{0}^{x}\frac{\frac{1}{1-k^{2}\beta^{2}}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t
+\int_{\frac{1}{k}}^{\frac{1}{k}\sqrt{\frac{1-k^{2}x^{2}}{1-x^{2}}}}\frac{\frac{(1-k^{2})\beta^{2}}{(1-\beta^{2})(1-k^{2}\beta^{2})}}{\left(1-\left(\frac{1-k^{2}\beta^{2}}{1-\beta^{2}}\right)t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}{\mathrm{d}}t
\end{align*}

青青子衿

(两实两虚)


\begin{gather*}
\int_{\beta}^{+\infty}\frac{1}{\sqrt{(t-\alpha)(t-\beta)((t-p)^{2}+q^{2})}}{\mathrm{d}}t\\
\\
=\int_{0}^{\frac{2(\sqrt{(p-\alpha)^{2}+q^{2}}-\sqrt{(p-\beta)^{2}+q^{2}})\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}{(\beta-\alpha)(2p-\alpha-\beta)}}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}({\scriptsize1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}})t^{2})}}{\mathrm{d}}t\\
\\
(\alpha<\beta)

\end{gather*}

\begin{gather*}
\int_{\beta}^{x}\frac{1}{\sqrt{(t-\alpha)(t-\beta)((t-p)^{2}+q^{2})}}{\mathrm{d}}t\\
\\
=\int_{0}^{\frac{2\sqrt{\scriptsize(x-\alpha)(x-\beta)\sqrt{((\alpha-p)^{2}+q^{2})((\beta-p)^{2}+q^{2})}}}
{\small(x-\beta)\sqrt{(\alpha-p)^{2}+q^{2}}+(x-\alpha)\sqrt{(\beta-p)^{2}+q^{2}}}}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}({\scriptsize1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}})t^{2})}}{\mathrm{d}}t\\
=\int_{0}^{
\sqrt{\Tiny1-\left(\frac{(\alpha-x)\sqrt{(\beta-p)^{2}+q^{2}}-(\beta-x)\sqrt{(\alpha-p)^{2}+q^{2}}}{(\alpha-x)\sqrt{(\beta-p)^{2}+q^{2}}+(\beta-x)\sqrt{(\alpha-p)^{2}+q^{2}}}\right)^{2}}
}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}({\scriptsize1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}})t^{2})}}{\mathrm{d}}t\\
\\
(\alpha<\beta<x)
\end{gather*}

\begin{gather*}
\int_{x}^{+\infty}\frac{1}{\sqrt{(t-\alpha)(t-\beta)((t-p)^{2}+q^{2})}}{\mathrm{d}}t\\
\\
=\int_{0}^{f}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}({\scriptsize1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}})t^{2})}}{\mathrm{d}}t\\
\\
f={\Tiny\tfrac{2(p^{2}+q^{2}-\alpha\beta-(2p-\alpha-\beta)x)((p-\alpha)^{2}+q^{2})^{1/4}((p-\beta)^{2}+q^{2})^{1/4}}{\sqrt{(x-\alpha)(x-\beta)}\left(p^{2}+q^{2}-\alpha\beta-\sqrt{((p-\alpha)^{2}+q^{2})((p-\beta)^{2}+q^{2})}-(2p-\alpha-\beta)x\right)+\sqrt{(p-x)^{2}+q^{2}}\left(p^{2}+q^{2}-\alpha\beta+\sqrt{((p-\alpha)^{2}+q^{2})((p-\beta)^{2}+q^{2})}-(2p-\alpha-\beta)x\right)}}\\
\\
(\alpha<\beta<x)

\end{gather*}




(2 Sqrt[Sqrt[((\[Alpha] - p)^2 + q^2) ((\[Beta] - p)^2 +
      q^2)] (x - \[Alpha]) (x - \[Beta])])/(
Sqrt[(\[Alpha] - p)^2 + q^2] (x - \[Beta]) +
  Sqrt[(\[Beta] - p)^2 + q^2] (x - \[Alpha])) /. {\[Alpha] ->
   1, \[Beta] -> 4, p -> 1, q -> 3}
(D[%, x]/Sqrt[Sqrt[((\[Alpha] - p)^2 + q^2) ((\[Beta] - p)^2 + q^2)]]/
   Sqrt[(1 - (%)^2) (1 - (
       1 + ((p - \[Alpha]) (p - \[Beta]) + q^2)/
        Sqrt[((\[Alpha] - p)^2 + q^2) ((\[Beta] - p)^2 + q^2)])/
       2  (%)^2)])^2 /. {\[Alpha] -> 1, \[Beta] -> 4, p -> 1,
   q -> 3} // Factor
1/( (x^2 - (\[Alpha] + \[Beta]) x + \[Alpha]*\[Beta]) (q^2 + 2 p*x +
     x^2)) /. {\[Alpha] -> 1, \[Beta] -> 4, p -> 1, q -> 3} // Factor

2. \int_{\beta}^{x}\frac{1}{\sqrt{(t-\alpha)(t-\beta)((t-p)^{2}+q^{2})}}dt
3. \int_{0}^{\frac{2\sqrt{(x-\alpha)(x-\beta)\sqrt{\left((\alpha-p)^{2}+q^{2}\right)\left((\beta-p)^{2}+q^{2}\right)}}}{(x-\beta)\sqrt{(\alpha-p)^{2}+q^{2}}+(x-\alpha)\sqrt{(\beta-p)^{2}+q^{2}}}}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}\left(1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}\right)t^{2})}}dt
4. \int_{0}^{\frac{2(\sqrt{(p-\alpha)^{2}+q^{2}}-\sqrt{(p-\beta)^{2}+q^{2}})\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}{(\beta-\alpha)(2p-\alpha-\beta)}}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}\left(1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}\right)t^{2})}}dt-\int_{0}^{\frac{2\sqrt{(x-\alpha)(x-\beta)\sqrt{\left((\alpha-p)^{2}+q^{2}\right)\left((\beta-p)^{2}+q^{2}\right)}}}{(x-\beta)\sqrt{(\alpha-p)^{2}+q^{2}}+(x-\alpha)\sqrt{(\beta-p)^{2}+q^{2}}}}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}\left(1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}\right)t^{2})}}dt
5. \int_{0}^{\frac{2(p^{2}+q^{2}-\alpha\beta-x(2p-\alpha-\beta))((p-\alpha)^{2}+q^{2})^{1/4}((p-\beta)^{2}+q^{2})^{1/4}}{\sqrt{(x-\alpha)(x-\beta)}\left(p^{2}+q^{2}-\alpha\beta-\sqrt{((p-\alpha)^{2}+q^{2})((p-\beta)^{2}+q^{2})}-x(-\alpha+2p-\beta)\right)+\sqrt{(p-x)^{2}+q^{2}}\left(p^{2}+q^{2}-\alpha\beta+\sqrt{((p-\alpha)^{2}+q^{2})((p-\beta)^{2}+q^{2})}-x(2p-\alpha-\beta)\right)}}\frac{\frac{1}{\left((p-\alpha)^{2}+q^{2}\right)^{1/4}\left((p-\beta)^{2}+q^{2}\right)^{1/4}}}{\sqrt{(1-t^{2})(1-\frac{1}{2}\left(1+\frac{(p-\alpha)(p-\beta)+q^{2}}{\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}\right)t^{2})}}dt
6. \frac{1}{q}\int_{\frac{1}{\sqrt{\frac{(p-\alpha)(p-\beta)+q^{2}+\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}{2q^{2}}}}}^{\frac{\sqrt{\frac{(x-\alpha)(x-\beta)}{(x-p)^{2}+q^{2}}}}{\sqrt{\frac{(p-\alpha)(p-\beta)+q^{2}+\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}{2q^{2}}}}}\frac{\frac{1}{\sqrt{-\frac{(p-\alpha)(p-\beta)+q^{2}-\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}{2q^{2}}}}}{\sqrt{(1-t^{2})(1-\frac{\frac{(p-\alpha)(p-\beta)+q^{2}+\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}{2q^{2}}}{\frac{(p-\alpha)(p-\beta)+q^{2}-\sqrt{\left((p-\alpha)^{2}+q^{2}\right)\left((p-\beta)^{2}+q^{2}\right)}}{2q^{2}}}t^{2})}}dt

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青青子衿

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

\\

\int_{0}^{x}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}{\mathrm{d}}t
&=x\sqrt{\frac{1-k^{2}x^{2}}{1-x^{2}}}-
\int_{\frac{1}{k}}^{\frac{1}{k}\sqrt{\frac{1-k^{2}x^{2}}{1-x^{2}}}}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}{\mathrm{d}}t\\
\\
&\qquad\quad\>\>(-1<x<1;0<k<1)

\end{align*}

1. \int_{\frac{1}{k}}^{\frac{1}{k}\sqrt{\frac{1-k^{2}x^{2}}{1-x^{2}}}}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt
2. x\sqrt{\frac{1-k^{2}x^{2}}{1-x^{2}}}-\int_{0}^{x}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt

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青青子衿

\begin{gather*}
\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{((t-a)^{2}+b^{2})((t-c)^{2}+d^{2})}}\\
\\
=\int_{\sqrt{\frac{1}{2} \left(1+\frac{(a-c)^2-b^2+d^2}{A B}\right)}}^{\frac{\sqrt{\frac{1}{2} \left(1+\frac{(a-c)^2-b^2+d^2}{A B}\right)} \left(x+\frac{a^2+b^2-c^2-d^2+A B}{2 (c-a)}\right)}{\sqrt{(x-a)^2+b^2}}}\frac{\frac{A-B}{2 b d}}{\sqrt{(1-t^{2})(1-\frac{A B (A-B)^2}{4 b^2 d^2}t^{2})}}{\mathrm{d}}t\\
\\
=\int_{0}^{\frac{\frac{(A-B)((a-c)^{2}+AB+b^{2}+d^{2})}{2bd}\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}}{2(c-a)}+x\right)}{\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}+x\right)\sqrt{(x-a)^{2}+b^{2}}+\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}-AB}{2(c-a)}+x\right)\sqrt{(x-c)^{2}+d^{2}}}}\frac{\frac{A-B}{2 b d}}{\sqrt{(1-t^{2})(1-\frac{A B (A-B)^2}{4 b^2 d^2}t^{2})}}{\mathrm{d}}t\\
\\
(a<c)\\
\\
\left\{
\begin{split}
A&=\sqrt{(a-c)^2+(b+d)^2}\\
B&=\sqrt{(a-c)^2+(b-d)^2}
\end{split}
\right.
\end{gather*}

1. \int_{x}^{+\infty}\frac{1}{\sqrt{((t-a)^{2}+b^{2})((t-c)^{2}+d^{2})}}dt
2. A=\sqrt{(a-c)^{2}+(b+d)^{2}}
3. B=\sqrt{(a-c)^{2}+(b-d)^{2}}
4. \int_{\frac{\sqrt{\frac{1}{2}(1+\frac{(a-c)^{2}-b^{2}+d^{2}}{AB})}\left(x+\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}\right)}{\sqrt{(x-a)^{2}+b^{2}}}}^{\sqrt{\frac{1}{2}(1+\frac{(a-c)^{2}-b^{2}+d^{2}}{AB})}}\frac{\frac{A-B}{2bd}}{\sqrt{(1-t^{2})(1-\frac{AB(A-B)^{2}}{4b^{2}d^{2}}t^{2})}}dt
5. \int_{0}^{\frac{\frac{(A-B)((a-c)^{2}+AB+b^{2}+d^{2})}{2bd}\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}}{2(c-a)}+x\right)}{\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}+x\right)\sqrt{(x-a)^{2}+b^{2}}+\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}-AB}{2(c-a)}+x\right)\sqrt{(x-c)^{2}+d^{2}}}}\frac{\frac{A-B}{2bd}}{\sqrt{(1-t^{2})(1-\frac{AB(A-B)^{2}}{4b^{2}d^{2}}t^{2})}}dt

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\begin{gather*}
\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{((t-a)^{2}+b^{2})((t-c)^{2}+d^{2})}}\\
=\int_{0}^{H(x)}\frac{\frac{A-B}{2bd}}{\sqrt{(1-t^{2})(1-\frac{AB(A-B)^{2}}{4b^{2}d^{2}}t^{2})}}{\mathrm{d}}t\\
\\
H(x)={\scriptsize\tfrac{\frac{(A-B)(a^{2}+b^{2}-c^{2}-d^{2})\left((a-c)^{2}+AB+b^{2}+d^{2}\right)}{2bd}x\left(x-\frac{2(a^{2}c+b^{2}c-ac^{2}-ad^{2})}{a^{2}+b^{2}-c^{2}-d^{2}}\right)}{\left(a^{2}+b^{2}-c^{2}-d^{2}+AB\right)\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}-AB}{2(c-a)}+x\right)\sqrt{\left(a^{2}+b^{2}\right)\left((x-c)^{2}+d^{2}\right)}+\left(a^{2}+b^{2}-c^{2}-d^{2}-AB\right)\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}+x\right)\sqrt{\left(c^{2}+d^{2}\right)\left((x-a)^{2}+b^{2}\right)}}}
\end{gather*}

1. T=\frac{\frac{(A-B)(a^{2}+b^{2}-c^{2}-d^{2})\left((a-c)^{2}+AB+b^{2}+d^{2}\right)}{2bd}x\left(x-\frac{2(a^{2}c+b^{2}c-ac^{2}-ad^{2})}{a^{2}+b^{2}-c^{2}-d^{2}}\right)}{\left(a^{2}+b^{2}-c^{2}-d^{2}+AB\right)\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}-AB}{2(c-a)}+x\right)\sqrt{\left(a^{2}+b^{2}\right)\left((x-c)^{2}+d^{2}\right)}+\left(a^{2}+b^{2}-c^{2}-d^{2}-AB\right)\left(\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}+x\right)\sqrt{\left(c^{2}+d^{2}\right)\left((x-a)^{2}+b^{2}\right)}}

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\begin{gather*}
\int_{a-bg}^{x}\frac{{\mathrm{d}}t}{\sqrt{((t-a)^{2}+b^{2})((t-c)^{2}+d^{2})}}\\
=\int_{0}^{\frac{x-a+bg}{\sqrt{(1+g^{2})((x-a)^{2}+b^{2})}}}\frac{\frac{A-B}{2bd}}{\sqrt{(1-t^{2})(1-\frac{AB(A-B)^{2}}{4b^{2}d^{2}}t^{2})}}{\mathrm{d}}t\\
\\
g=\frac{4b^{2}-(A-B)^{2}}{4(c-a)b}
\end{gather*}


\begin{gather*}
\int_{-\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}}^{x}\frac{{\mathrm{d}}t}{\sqrt{((t-a)^{2}+b^{2})((t-c)^{2}+d^{2})}}\\
=\int_{0}^{\frac{\sqrt{\frac{1}{2}(1+\frac{(a-c)^{2}-b^{2}+d^{2}}{AB})}\left(x+\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}\right)}{\sqrt{(x-a)^{2}+b^{2}}}}\frac{\frac{A-B}{2bd}}{\sqrt{(1-t^{2})(1-\frac{AB(A-B)^{2}}{4b^{2}d^{2}}t^{2})}}{\mathrm{d}}t\\
\\
\left\{
\begin{split}
A&=\sqrt{(a-c)^2+(b+d)^2}\\
B&=\sqrt{(a-c)^2+(b-d)^2}
\end{split}
\right.

\end{gather*}

青青子衿

青青子衿 发表于 2024-5-6 04:31
\begin{gather*}
\int_{-\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}}^{x}\frac{{\mathrm{d}}t}{\sqrt{((t-a)^{2}+b^{2})((t-c)^{2}+d^{2})}}\\
=\int_{0}^{\frac{\sqrt{\frac{1}{2}(1+\frac{(a-c)^{2}-b^{2}+d^{2}}{AB})}\left(x+\frac{a^{2}+b^{2}-c^{2}-d^{2}+AB}{2(c-a)}\right)}{\sqrt{(x-a)^{2}+b^{2}}}}\frac{\frac{A-B}{2bd}}{\sqrt{(1-t^{2})(1-\frac{AB(A-B)^{2}}{4b^{2}d^{2}}t^{2})}}{\mathrm{d}}t\\
\\
\left\{
\begin{split}
A&=\sqrt{(a-c)^2+(b+d)^2}\\
B&=\sqrt{(a-c)^2+(b-d)^2}
\end{split}
\right.

\end{gather*}

\begin{gather*}
\int_{0}^{u+vi}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}\\

=\int_{0}^{\sqrt{U_1U_2}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}+i\int_{0}^{\sqrt{V_1V_2}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(1-k^{2})t^{2})}}\\
\\
\\
\left\{\begin{split}
U_{1}&=\frac{1+u^{2}+v^{2}-\Delta_{1}}{2ku}\\
U_{2}&=\frac{1+k^{2}(u^{2}+v^{2})-\Delta_{2}}{2ku}\\
V_{1}&=1-\frac{\Delta_{1}-(u^{2}+v^{2})\Delta_{2}}{1-4u^{2}+(u^{2}+v^{2})(2+k^{2}(u^{2}+v^{2}))}\\
V_{2}&=\frac{(1+u^{2}+v^{2})(u^{2}+v^{2})}{4u^{2}}-\frac{\Delta_{1}^{2}-\Delta_{1}\Delta_{2}}{4(1-k^{2})u^{2}}\\
\Delta_1&=\sqrt{((1-u)^{2}+v^{2})((1+u)^{2}+v^{2})}\\
\Delta_{2}&=\sqrt{((1-ku)^{2}+k^{2}v^{2})((1+ku)^{2}+k^{2}v^{2})}
\end{split}\right.
\end{gather*}

青青子衿

青青子衿 发表于 2023-6-2 03:01
\begin{align*}
\operatorname{sn}\!\left(\xi,k\mathrm{i}\right)
&=\frac{1}{ \sqrt{1+k^2}}\operatorname{sd}\left(\sqrt{1+k^2} \,\xi,\frac{k}{\sqrt{1+k^2}}\right)\\
\\
\operatorname{sn}^{-1}\!\left(x,k\mathrm{i}\right)&=\frac{1}{ \sqrt{1+k^2}}\operatorname{sn}^{-1}\!\left(x\sqrt{\frac{1+k^2}{1+k^2x^2}},\frac{k}{\sqrt{1+k^2}}\right)\\
\end{align*}

【虚模变换】

\begin{align*}
\dfrac{1}{i}\int_{0}^{ix}\frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-(l t)^2)}}&=\int_{0}^{\frac{x}{\sqrt{1+x^2}}}\frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-(1-l^2)t^2)}}\\

\dfrac{1}{i}\int_{0}^{\frac{ix}{\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(lt)^{2})}}&=\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(1-l^2)t^{2})}}
\end{align*}

1. NIntegrate[
2. 1/(I Sqrt[(1 - t^2) (1 - (l*t)^2)]) /. l -> 0.1624, {t, 0,
3.   I*y /. {y -> 0.232, l -> 0.1624}}]
4. NIntegrate[
5. 1/Sqrt[(1 + t^2) (1 + (l*t)^2)] /. l -> 0.1624, {t, 0,
6.   y /. {y -> 0.232, l -> 0.1624}}]
7. NIntegrate[
8. 1/Sqrt[(1 - t^2) (1 - (Sqrt[1 - l^2] t)^2)] /. l -> 0.1624, {t, 0,
9.   y/Sqrt[1 + y^2] /. {y -> 0.232, l -> 0.1624}}]

复制代码

\begin{gather*}
{\Large{\int}}_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\,t)^{2})}}\\
\\
=\frac{1}{i\sqrt{7}}{\Large{\int}}_{0}^{\frac{ix(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(3-\sqrt{7})x^{2}+4(69-26\sqrt{7})x^{4}-(127-48\sqrt{7})x^{6})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\,t)^{2})}}
\end{gather*}

\begin{gather*}
2\sqrt[4]{7}{\large{\int}}_{\scriptsize\frac{(7-4\sqrt{7})x^{2}+8\sqrt{7}+8\sqrt{7(1-x^{2})}}{x^{2}}}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{3}-35t-98}}\\
={\large{\int}}_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\raise{0.8pt}{\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\,t)^{2})}}\\
\end{gather*}

\begin{gather*}
\sqrt{7}\int_{-\infty}^{x}\frac{{\mathrm{d}}t}{\sqrt{98+35t-t^{3}}}\\
=\int_{\scriptsize{-\frac{x}{7}-\frac{112(3x^{5}-x^{4}-210x^{3}-1106x^{2}-2625x-3773)}{7(x^{3}+7x^{2}-21x-91)^{2}}}}^{+\infty}\frac{1}{\sqrt{t^{3}-35t-98}}dt\\
\end{gather*}

1. ((I Sqrt[7])/Sqrt[
2.    u^3 - 35 u -
3.     98])^2 - (D[-(u/7) - (
4.      112 (3 u^5 - u^4 - 210 u^3 - 1106 u^2 - 2625 u - 3773))/(
5.      7 (u^3 + 7 u^2 - 21 u - 91)^2), u]/
6.    Sqrt[(-(u/7) - (
7.       112 (3 u^5 - u^4 - 210 u^3 - 1106 u^2 - 2625 u - 3773))/(
8.       7 (u^3 + 7 u^2 - 21 u - 91)^2))^3 -
9.     35 (-(u/7) - (
10.        112 (3 u^5 - u^4 - 210 u^3 - 1106 u^2 - 2625 u - 3773))/(
11.        7 (u^3 + 7 u^2 - 21 u - 91)^2)) - 98])^2 // Factor
12. Y^2 - (X^3 - 35 X - 98) /. {X -> -(u/7) - (
13.      112 (3 u^5 - u^4 - 210 u^3 - 1106 u^2 - 2625 u - 3773))/(
14.      7 (u^3 + 7 u^2 - 21 u - 91)^2),
15.    Y -> v/(7 Sqrt[7]
16.        I) (1 - (
17.        112 (3 u^7 - 23 u^6 - 441 u^5 - 4571 u^4 - 33383 u^3 -
18.           135093 u^2 - 251811 u - 80409))/(u^3 + 7 u^2 - 21 u -
19.          91)^3)} // Factor
21. Y^2 - (X^3 - 35 X + 98) /. {X ->
22.     1/(Sqrt[7] I)^2 (u + (
23.        112 (3 u^5 + u^4 - 210 u^3 + 1106 u^2 - 2625 u + 3773))/(u^3 -
24.          7 u^2 - 21 u + 91)^2),
25.    Y -> v/(Sqrt[7] I)^3 (1 - (
26.        112 (3 u^7 + 23 u^6 - 441 u^5 + 4571 u^4 - 33383 u^3 +
27.           135093 u^2 - 251811 u + 80409))/(u^3 - 7 u^2 - 21 u +
28.          91)^3)} // Factor

复制代码

\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{align*}
\int_0^x\frac{{\mathrm{d}}t}{\sqrt{(1-t^2)(1-({\raise{0.4pt}\scriptsize{\frac{\sqrt{6}-\sqrt{2}}{4}}})^2t^2)}}
=
\frac{1}{\sqrt{-3\,}}\int_0^{\frac{\sqrt{-3}\,x \left(1+{\scriptsize\frac{2 \sqrt{3}-3}{6}} x^2\right)}{(1-{\scriptsize\frac{2-\sqrt{3}}{2}}x^2)\sqrt{1-x^2}}}

\frac{{\mathrm{d}}t}{\sqrt{(1-t^2)(1-({\raise{0.4pt}\scriptsize{\frac{\sqrt{6}-\sqrt{2}}{4}}})^2t^2)}}\\

\end{align*}


\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-\frac{s^{3}(2+s)}{1+2s}t^{2})}}dt\\
=\frac{1}{1+2s}\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1+t^{2})(1+\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}\\
=\frac{1}{(1+2s)i}\int_{0}^{\frac{ix(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}\\
\end{gather*}

1. \int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-\frac{s^{3}(2+s)}{1+2s}t^{2})}}dt
2. s=\frac{\sqrt{3}-1}{2}
3. \frac{1}{1+2s}\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{1}{\sqrt{(1+t^{2})(1+\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}dt

复制代码

青青子衿

青青子衿 发表于 2024-8-21 22:26
\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-\frac{s^{3}(2+s)}{1+2s}t^{2})}}dt\\
=\frac{1}{1+2s}\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1+t^{2})(1+\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}\\
=\frac{1}{(1+2s)i}\int_{0}^{\frac{ix(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}\\
\end{gather*}

\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-\frac{s^{3}(2+s)}{1+2s}t^{2})}}dt\\

=\frac{1}{(1+2s)i}\int_{0}^{\frac{ix(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}\\
\end{gather*}



\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&=\tfrac{s^{3}(2+s)}{1+2s}\\
V&=\tfrac{(1-s)^{3}(1+s)}{(1+2s)^{3}}\\
y&=\tfrac{ix(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}\\
\frac{\partial\,y}{\partial\,x}&=\tfrac{i\ (1+2sx^{2}+s^{2}x^{2})(1+2s-2sx^{2}-s^{2}x^{2})}{(1-s^{2}x^{2})^{2}(1-x^{2})^{3/2}}\\
\frac{\partial\,y}{\partial\,s}&=\tfrac{2i\>\!x\>\!(1+2sx^{2}+s^{2}x^{2})}{(1-s^{2}x^{2})^{2}\sqrt{1-x^{2}}}\\


M(s)&=\tfrac{1}{(1+2s)i}\\
N(s)&={\small{(1+2s)i}}\\
\\

\\
\Phi(x,U)&=M(s)\Phi(y,V)\\
\Phi(y,V)&=\frac{\Phi(x,U)}{M(s)}=N(s)\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}\\

\\
\end{align*}



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

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

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

\end{align*}

\begin{align*}
\Omega&=\frac{\partial\,\Phi(y,V)}{\partial\,s}-\frac{\partial}{\partial\,s}\left(\frac{\Phi(x,U)}{M(s)}\right)\\

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

\end{align*}


\begin{align*}
\Omega^*&=\frac{\Omega}{\frac{\frac{\partial\,V}{\partial\,s}}{2V\left(1-V\right)}}=\frac{2V\left(1-V\right)}{\frac{\partial\,V}{\partial\,s}}\left(\frac{\partial\,\Phi(y,V)}{\partial\,s}-\frac{\partial}{\partial\,s}\left(\frac{\Phi(x,U)}{M(s)}\right)\\
\right)\\
&=\mathcal{E}(y,V)-\frac{\frac{V\left(1-V\right)\frac{\partial\,U}{\partial\,s}}{U\left(1-U\right)\frac{\partial\,V}{\partial\,s}}}{M}\cdot\mathcal{E}(x,U)\\
&\qquad\quad-\frac{2V\left(1-V\right)}{\frac{\partial\,V}{\partial\,s}}\left(\frac{\partial\big(\frac{1}{M(s)}\big)}{\partial\,s}-\frac{\frac{\partial\,U}{\partial\,s}}{2UM}+\dfrac{\frac{\partial\,V}{\partial\,s}}{2VM}\right)\cdot\Phi(x,U)\\

&\qquad\qquad+\dfrac{2V\left(1-V\right)\cdot\frac{\partial\,\Phi(x,U)}{\partial\,x}}{\frac{\partial\,V}{\partial\,s}\cdot\,\!M}\cdot\left(\frac{x(1-x^2)\frac{\partial\,U}{\partial\,s}}{2(1-U)}+\frac{\frac{\partial\,y}{\partial\,s}-\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,V}{\partial\,s}}{\frac{\partial\,\!y}{\partial\,\!x}}\right)\\

\\
\Omega^*&=\mathcal{E}(y,V)-\mathfrak{n}M\cdot\mathcal{E}(x,U)\\
&\qquad\quad-\frac{2V\left(1-V\right)}{\frac{\partial\,V}{\partial\,s}}\left(\frac{\partial\big(\frac{1}{M(s)}\big)}{\partial\,s}-\frac{\frac{\partial\,U}{\partial\,s}}{2UM}+\dfrac{\frac{\partial\,V}{\partial\,s}}{2VM}\right)\cdot\Phi(x,U)\\

&\qquad\qquad+\dfrac{2V\left(1-V\right)}{\frac{\partial\,V}{\partial\,s}\cdot\,\!M\sqrt{(1-x^2)(1-Ux^2)}}\cdot\left(\frac{x(1-x^2)\frac{\partial\,U}{\partial\,s}}{2(1-U)}+\frac{\frac{\partial\,y}{\partial\,s}-\frac{y(1-y^2)}{2(1-V)}\cdot\frac{\partial\,V}{\partial\,s}}{\frac{\partial\,\!y}{\partial\,\!x}}\right)\\
\end{align*}


\begin{align*}
&\qquad\qquad\quad\int_{0}^{\frac{ix(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\sqrt{\frac{1-{\raise{0.5pt}\scriptsize\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}}t^{2}}{1-t^{2}}}{\mathrm{d}}t\\

&=\frac{3}{(1+2s)i}\int_{0}^{x}\sqrt{\frac{1-{\raise{0.5pt}\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}}{1-t^{2}}}\mathrm{d}t\\

&\qquad+\frac{i\,(3+4s+2s^{2})}{1+2s}\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)(1-{\raise{0.5pt}\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2})}}\\

&\qquad\qquad+\frac{i\>\!x(1+2s^{2}-3s^{2}x^{2})}{(1+2s)(1-s^{2}x^{2})}\sqrt{\frac{1-{\raise{0.5pt}\scriptsize\frac{s^{3}(2+s)}{1+2s}}x^{2}}{1-x^{2}}}
\end{align*}



kuing.cjhb.site/forum.php?mod=viewthread&tid=11938

MMA关于变上限积分求导验证法

1. (D[Inactive[Integrate][1/
2.      Sqrt[(1 - t^2) (1 - ((1 - s)^3 (1 + s))/(1 + 2 s)^3 t^2)], {t,
3.       0, (I*\[Beta] (1 + 2 s + s^2 \[Beta]^2))/((1 -
4.          s^2 \[Beta]^2) Sqrt[1 - \[Beta]^2])}], \[Beta]])^2 -
5.   (D[Inactive[Integrate][((1 + 2 s) I)/
6.      Sqrt[(1 - t^2) (1 - (s^3 (2 + s))/(1 + 2 s) t^2)], {t,
7.       0, \[Beta]}], \[Beta]])^2 // Factor
8. FullSimplify[(D[
9.     Inactive[Integrate][Sqrt[(
10.      1 - ((1 - s)^3 (1 + s))/(1 + 2 s)^3*t^2)/(
11.      1 - t^2)], {t, 0, (
12.       I*\[Beta] (1 + 2 s + s^2 \[Beta]^2))/((1 - s^2 \[Beta]^2) Sqrt[
13.        1 - \[Beta]^2])}], \[Beta]])^2 -
14.   (D[Inactive[Integrate][
15.       3/((1 + 2 s) I) Sqrt[(1 - (s^3 (2 + s))/(1 + 2 s)*t^2)/(
16.         1 - t^2)] + (I (3 + 4 s + 2 s^2))/(1 + 2 s)/
17.        Sqrt[(1 - t^2) (1 - (s^3 (2 + s))/(1 + 2 s) t^2)], {t,
18.        0, \[Beta]}] + (
19.       I*\[Beta]*(1 + 2 s^2 - 3 s^2 \[Beta]^2) )/((1 + 2 s) (1 -
20.          s^2 \[Beta]^2) ) Sqrt[(
21.       1 - (s^3 (2 + s))/(1 + 2 s)*\[Beta]^2)/(
22.       1 - \[Beta]^2)], \[Beta]])^2,
23. Assumptions -> {1 - \[Beta]^2 > 0,
24.    1 - (s^3 (2 + s) \[Beta]^2)/(1 + 2 s) > 0}]

复制代码

青青子衿

青青子衿 发表于 2024-11-15 07:20
\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-\frac{s^{3}(2+s)}{1+2s}t^{2})}}dt\\

=\frac{1}{(1+2s)i}\int_{0}^{\frac{ix(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2})}}\\
\end{gather*}

\begin{align*}
&\qquad\qquad\qquad\int_{0}^{x}\frac{{\mathrm{d}}t}{\left(1-(1-k^{2})\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-(1-k^{2})t^{2}\right)}}\\
&=\int_{0}^{\frac{x}{\sqrt{1-x^{2}}}}\frac{\frac{1}{1-\beta^{2}}}{\left(1-\frac{k^{2}\beta^{2}}{1-\beta^{2}}t^{2}\right)\sqrt{\left(1+t^{2}\right)\left(1+k^{2}t^{2}\right)}}{\mathrm{d}}t\\

&\qquad-\frac{\beta}{\sqrt{(1-\beta^{2})(1-(1-k^{2})\beta^{2})}}\operatorname{arctanh}\left(\beta x\sqrt{\frac{(1-(1-k^{2})\beta^{2})(1-(1-k^{2})x^{2})}{(1-\beta^{2})(1-x^{2})}}\right)
\end{align*}

1. \int_{0}^{x}\frac{1}{\left(1-(1-k^{2})\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-(1-k^{2})t^{2}\right)}}dt
2. \int_{0}^{\frac{x}{\sqrt{1-x^{2}}}}\frac{\frac{1}{1-\beta^{2}}}{\left(1-\frac{k^{2}\beta^{2}}{1-\beta^{2}}t^{2}\right)\sqrt{\left(1+t^{2}\right)\left(1+k^{2}t^{2}\right)}}dt-\frac{\beta}{\sqrt{(1-\beta^{2})(1-(1-k^{2})\beta^{2})}}\operatorname{arctanh}\left(\beta x\sqrt{\frac{(1-(1-k^{2})\beta^{2})(1-(1-k^{2})x^{2})}{(1-\beta^{2})(1-x^{2})}}\right)

复制代码

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

&\quad\>\>-\int_{0}^{x}\left(\frac{3(1+\frac{s^{2}}{1+2s}\beta^{2})}{\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}-\frac{2}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}\right){\mathrm{d}}t\\

&=\frac{(1+\frac{s^{2}}{1+2s}\beta^{2})\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}\right)}}\operatorname{artanh}\left(\varPsi(x)\sqrt{\frac{(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2})(1-\frac{s^{3}(2+s)}{1+2s}x^{2})}{(1-\beta^{2})(1-x^{2})}}\right)\\
\\
\\
&\qquad\varPsi(x)=\tfrac{\left(1+2s^{2}-3\beta^{2}s^{2}\right)\beta x\left(1-\frac{s^{2}\left(3+6s-4\beta^{2}-4s\beta^{2}+2s^{2}\beta^{2}-4s^{3}\beta^{2}-2s^{4}\beta^{2}+2s^{3}\beta^{4}+s^{4}\beta^{4}\right)}{(1+2s)(1+2s^{2}-3\beta^{2}s^{2})}x^{2}-\frac{\beta^{2}s^{5}(2+s)(1+2s-2s\beta^{2}-s^{2}\beta^{2})}{(1+2s)^{2}(1+2s^{2}-3\beta^{2}s^{2})}x^{4}\right)}{\left(1-\beta^{2}s^{2}\right)\left(1-\frac{s^{2}\left(1+2s-4\beta^{2}-4s\beta^{2}+2s^{2}\beta^{2}-4s^{3}\beta^{2}-2s^{4}\beta^{2}+6s^{3}\beta^{4}+3s^{4}\beta^{4}\right)}{(1+2s)(1-\beta^{2}s^{2})}x^{2}-\frac{s^{5}\beta^{2}(s+2)\left(3+6s-2s\beta^{2}-s^{2}\beta^{2}-4s^{3}\beta^{2}-2s^{4}\beta^{2}\right)}{(1+2s)^{2}(1-\beta^{2}s^{2})}x^{4}\right)}
\end{align*}

1. \int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{(1-s^{2}x^{2})\sqrt{1-x^{2}}}}\frac{\frac{(1+s(s+2)\beta^{2})(1-\frac{s(2+s)}{1+2s}\beta^{2})}{(1+2s)(1-\beta^{2})(1-\beta^{2}s^{2})}}{\left(1-\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}\frac{\beta^{2}(1+2s+s^{2}\beta^{2})^{2}}{(1-s^{2}\beta^{2})^{2}(1-\beta^{2})}t^{2}\right)\sqrt{\left(1+t^{2}\right)\left(1+\frac{(1-s)^{3}(1+s)}{(1+2s)^{3}}t^{2}\right)}}dt-\int_{0}^{x}\left(\frac{3(1+\frac{s^{2}}{1+2s}\beta^{2})}{\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}-\frac{2}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}\right)dt
2. \frac{(1+\frac{s^{2}}{1+2s}\beta^{2})\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{\left(1+2s^{2}-3\beta^{2}s^{2}\right)\beta x\left(1-\frac{s^{2}\left(3+6s-4\beta^{2}-4s\beta^{2}+2s^{2}\beta^{2}-4s^{3}\beta^{2}-2s^{4}\beta^{2}+2s^{3}\beta^{4}+s^{4}\beta^{4}\right)}{(1+2s)(1+2s^{2}-3\beta^{2}s^{2})}x^{2}-\frac{\beta^{2}s^{5}(2+s)(1+2s-2s\beta^{2}-s^{2}\beta^{2})}{(1+2s)^{2}(1+2s^{2}-3\beta^{2}s^{2})}x^{4}\right)}{\left(1-\beta^{2}s^{2}\right)\left(1-\frac{s^{2}\left(1+2s-4\beta^{2}-4s\beta^{2}+2s^{2}\beta^{2}-4s^{3}\beta^{2}-2s^{4}\beta^{2}+6s^{3}\beta^{4}+3s^{4}\beta^{4}\right)}{(1+2s)(1-\beta^{2}s^{2})}x^{2}-\frac{s^{5}\beta^{2}(s+2)\left(3+6s-2s\beta^{2}-s^{2}\beta^{2}-4s^{3}\beta^{2}-2s^{4}\beta^{2}\right)}{(1+2s)^{2}(1-\beta^{2}s^{2})}x^{4}\right)}\sqrt{\frac{(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2})(1-\frac{s^{3}(2+s)}{1+2s}x^{2})}{(1-\beta^{2})(1-x^{2})}}\right)

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青青子衿

\begin{gather*}
\int_{0}^{1}\frac{{\mathrm{d}t}}{\sqrt{(1-t^{2})(1-Ut^{2})}}=5\int_{0}^{\beta}\frac{{\mathrm{d}t}}{\sqrt{(1-t^{2})(1-Ut^{2})}}\\
\\
\int_{0}^{1}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}{\mathrm{d}t}=5\int_{0}^{\beta}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}{\mathrm{d}t}+\tfrac{(3+s)(s^{2}-4s-1)}{4s(s^{3}+s^{2}-s)^{1/4}}\\
\\
\int_{0}^{1}\frac{{\mathrm{d}t}}{(1-U\beta^{2}t^{2})\sqrt{(1-t^{2})(1-Ut^{2})}}=\int_{0}^{1}\frac{\chi}{\sqrt{(1-t^{2})(1-Ut^{2})}}{\mathrm{d}t}\\
\\
\\
\left\{\begin{split}
\beta&=\tfrac{2s^{2}-2\sqrt{s^{3}+s^{2}-s}}{(1+s)(1-s)^{2}}\\
U&=\tfrac{1}{2}-\tfrac{1+2s-5s^{2}-5s^{4}-2s^{5}+s^{6}}{16s^{2}\sqrt{s^{3}+s^{2}-s}}\\
\chi&=\tfrac{2(1-3s^{3}-2s^{4})-2s(3+s)\sqrt{s^{3}+s^{2}-s}}{5(1-s)(1+s)^{3}}
\end{split}\right.
\end{gather*}

1. \int_{0}^{1}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}dt-5\int_{0}^{\beta}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}dt-\frac{(1+8s)(1-14s+4s^{2})}{12s(1-s)\left(3s(1-s)(1+s-11s^{2})\right)^{1/4}}
2. \int_{0}^{1}\frac{1}{(1-U\beta^{2}t^{2})\sqrt{(1-t^{2})(1-Ut^{2})}}dt
3. \left(\int_{0}^{1}\frac{1}{\sqrt{(1-t^{2})(1-Ut^{2})}}dt\right)\left(1+\frac{5\int_{0}^{\beta}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}dt-\int_{0}^{1}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}dt}{\frac{5\sqrt{(1-\beta^{2})(1-U\beta^{2})}}{\beta}}\right)
4. \int_{0}^{1}\frac{\chi}{\sqrt{(1-t^{2})(1-Ut^{2})}}dt
5. \beta=\frac{6s\left((1-s)^{2}-\sqrt{3s(1-s)(1+s-11s^{2})}\right)}{(1+2s)(1-4s)^{2}}
6. U=\frac{1}{2}-\frac{\left(1-2s+10s^{2}\right)\left(1-10s-30s^{2}+140s^{3}-20s^{4}\right)\sqrt{3s(1-s)(1+s-11s^{2})}}{432s^{3}(1-s)^{3}(1+s-11s^{2})}
7. \chi=1-\frac{(1+8s)\left(1-30s^{2}+2s^{3}-2(1-s)\sqrt{3s(1-s)(1+s-11s^{2})}\right)}{5(1+2s)^{3}(1-4s)}
8. s=0.21
9. u=\frac{\frac{3}{s}-3}{9}
10. \frac{\frac{2u^{2}-2\sqrt{u^{3}+u^{2}-u}}{(1+u)(1-u)^{2}}}{\beta}
11. \frac{\frac{1}{2}-\frac{1+2u-5u^{2}-5u^{4}-2u^{5}+u^{6}}{16u^{2}\sqrt{u^{3}+u^{2}-u}}}{U}
12. \frac{\frac{2(1-3u^{3}-2u^{4})-2u(u+3)\sqrt{u^{3}+u^{2}-u}}{5(1-u)(u+1)^{3}}}{\chi}
13. \frac{\frac{(3+u)(u^{2}-4u-1)}{4u(u^{3}+u^{2}-u)^{1/4}}}{\frac{(1+8s)(1-14s+4s^{2})}{12s(1-s)\left(3s(1-s)(1+s-11s^{2})\right)^{1/4}}}

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1. \int_{0}^{1}\sqrt{\frac{1-Vt^{2}}{1-t^{2}}}dt-5\int_{0}^{b}\sqrt{\frac{1-Vt^{2}}{1-t^{2}}}dt-\frac{(3+u)(u^{2}-4u-1)}{4u(u^{3}+u^{2}-u)^{1/4}}
2. \int_{0}^{1}\frac{1}{(1-Vb^{2}t^{2})\sqrt{(1-t^{2})(1-Vt^{2})}}dt-\int_{0}^{1}\frac{X}{\sqrt{(1-t^{2})(1-Vt^{2})}}dt
3. V=\frac{1}{2}-\frac{1+2u-5u^{2}-5u^{4}-2u^{5}+u^{6}}{16u^{2}\sqrt{u^{3}+u^{2}-u}}
4. b=\frac{2u^{2}-2\sqrt{u^{3}+u^{2}-u}}{(1+u)(1-u)^{2}}
5. X=\frac{2(1-3u^{3}-2u^{4})-2u(u+3)\sqrt{u^{3}+u^{2}-u}}{5(1-u)(u+1)^{3}}
6. \int_{0}^{1}\frac{1}{(1-\left(\frac{1}{2}+\frac{1+3x-x^{2}-x^{3}}{4x\sqrt{x^{3}+x^{2}-x}}\right)t^{2})\sqrt{(1-t^{2})(1-\left(\frac{1}{2}-\frac{1+2x-5x^{2}-5x^{4}-2x^{5}+x^{6}}{16x^{2}\sqrt{x^{3}+x^{2}-x}}\right)t^{2})}}dt-\int_{0}^{1}\frac{\frac{2(1-3x^{3}-2x^{4})-2x(3+x)\sqrt{x^{3}+x^{2}-x}}{5(1-x)(1+x)^{3}}}{\sqrt{(1-t^{2})(1-\left(\frac{1}{2}-\frac{1+2x-5x^{2}-5x^{4}-2x^{5}+x^{6}}{16x^{2}\sqrt{x^{3}+x^{2}-x}}\right)t^{2})}}dt

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青青子衿

\begin{gather*}
\int_{0}^{1}\frac{{\mathrm{d}t}}{\sqrt{(1-t^{2})(1-Ut^{2})}}=\frac{5}{4}\int_{0}^{\beta}\frac{{\mathrm{d}t}}{\sqrt{(1-t^{2})(1-Ut^{2})}}\\
\\
\\
\\
\left\{\begin{split}
\beta&=\sqrt{\tfrac{5(461+438s-186s^{2}+2s^{3}+s^{4})+10(19-24s+s^{2})\sqrt{62-33s-s^{2}+s^{3}}}{8(31-s-s^{2})^{2}}
-\tfrac{\sqrt{5}(29-4s+s^{2})(19-24s+s^{2}-10\sqrt{62-33s-s^{2}+s^{3}})}{8(31-s-s^{2})^{2}}}\\
U&=\tfrac{640(2-s)(31-s-s^{2})^{5}}{(7-s)(3+s)(19-24s+s^{2})^{5}}
-\tfrac{16\sqrt{5}(29-4s+s^{2})(31-s-s^{2})^{2}(4661-862s-786s^{2}+102s^{3}+s^{4})\sqrt{62-33s-s^{2}+s^{3}}}{(7-s)(3+s)(19-24s+s^{2})^{5}}
\end{split}\right.
\end{gather*}

1. \beta\left(t\right)=\sqrt{\frac{5(461+438t-186t^{2}+2t^{3}+t^{4})+10(19-24t+t^{2})\sqrt{62-33t-t^{2}+t^{3}}}{8(31-t-t^{2})^{2}}-\frac{\sqrt{5}(29-4t+t^{2})(19-24t+t^{2}-10\sqrt{62-33t-t^{2}+t^{3}})}{8(31-t-t^{2})^{2}}}
2. U\left(t\right)=\frac{640(2-t)(31-t-t^{2})^{5}}{(7-t)(3+t)(19-24t+t^{2})^{5}}-\frac{16\sqrt{5}(29-4t+t^{2})(31-t-t^{2})^{2}(4661-862t-786t^{2}+102t^{3}+t^{4})\sqrt{62-33t-t^{2}+t^{3}}}{(7-t)(3+t)(19-24t+t^{2})^{5}}
3. \frac{\int_{0}^{\beta\left(s\right)}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-U\left(s\right)t^{2}\right)}}dt}{\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-U\left(s\right)t^{2}\right)}}dt}
4. s=1.6
5. x^{12}y^{6}-50x^{10}y^{5}+140x^{9}y^{5}+140x^{9}y^{4}-160x^{8}y^{5}-445x^{8}y^{4}-160x^{8}y^{3}+64x^{7}y^{5}+560x^{7}y^{4}+560x^{7}y^{3}+64x^{7}y^{2}-240x^{6}y^{4}-780x^{6}y^{3}-240x^{6}y^{2}+360x^{5}y^{3}+360x^{5}y^{2}-105x^{4}y^{2}-80x^{3}y^{2}-80x^{3}y+16x^{2}y^{2}+94x^{2}y+16x^{2}-20xy-20x+5=0
6. \left(\beta\left(t\right),U\left(t\right)\right)

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