后两类椭圆积分的加法

青青子衿

\begin{align*}
&\quad\phantom{=}\int_{0}^{u}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}{\mathrm{d}t}
+\int_{0}^{v}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}{\mathrm{d}t}\\
&=\int_{0}^{w}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}{\mathrm{d}t}-\frac{4A(A+\chi)uvw}{(u^{2}+A)(v^{2}+A)(w^{2}+A)}\\
&\qquad+\frac{u\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}}{u^{2}+A}+\frac{v\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}}{v^{2}+A}\\
&\qquad\qquad-\frac{w\sqrt{(w^{2}-\chi)^{2}+\psi^{2}}}{w^{2}+A}\\
&=\int_{0}^{w}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}{\mathrm{d}t}\\
&\qquad-\frac{u^{2}v\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}+v^{2}u\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}}{A^{2}-u^{2}v^{2}}\\
\\
&
\qquad\left\{\begin{split}
w&=\frac{Au\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}+Av\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}}{A^{2}-u^{2}v^{2}}\\
A&=\sqrt{\chi^{2}+\psi^{2}}
\end{split}\right.
\end{align*}

1. \int_{0}^{u}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt+\int_{0}^{v}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt
2. \int_{0}^{w}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt-\frac{4A(A+\chi)uvw}{(u^{2}+A)(v^{2}+A)(w^{2}+A)}+\frac{u\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}}{u^{2}+A}+\frac{v\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}}{v^{2}+A}-\frac{w\sqrt{(w^{2}-\chi)^{2}+\psi^{2}}}{w^{2}+A}
3. w=\frac{Au\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}+Av\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}}{A^{2}-u^{2}v^{2}}
4. A=\sqrt{\chi^{2}+\psi^{2}}
5. u=x
6. v=7.36
7. \chi=2
8. \psi=4.97
9. \int_{0}^{w}\frac{t^{2}}{\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt-\frac{u^{2}v\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}+v^{2}u\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}}{A^{2}-u^{2}v^{2}}
11. g\left(x\right)=-\frac{f\left(\frac{\frac{4A\left(A^{2}-\omega^{4}\right)\sqrt{\left(\omega^{2}-\chi\right)^{2}+\psi^{2}}}{\omega}uvw_{1}}{\left(\left(A+u^{2}\right)\left(A+v^{2}\right)\left(A+w_{1}^{2}\right)\left(A-\omega^{2}\right)^{2}+\left(A-u^{2}\right)\left(A-v^{2}\right)\left(A-w_{1}^{2}\right)\left(A+\omega^{2}\right)^{2}\right)}\right)}{2\omega\sqrt{\left(\omega^{2}-\chi\right)^{2}+\psi^{2}}}+\frac{f\left(\frac{u^{2}\left(\left(\omega^{2}-\chi\right)^{2}+\psi^{2}\right)+\omega^{2}\left(\left(u^{2}-\chi\right)^{2}+\psi^{2}\right)}{2u\omega\sqrt{\left(\left(u^{2}-\chi\right)^{2}+\psi^{2}\right)\left(\left(\omega^{2}-\chi\right)^{2}+\psi^{2}\right)}}\right)}{4\omega\sqrt{\left(\omega^{2}-\chi\right)^{2}+\psi^{2}}}+\frac{f\left(\frac{2v\omega\sqrt{\left(\left(v^{2}-\chi\right)^{2}+\psi^{2}\right)\left(\left(\omega^{2}-\chi\right)^{2}+\psi^{2}\right)}}{v^{2}\left(\left(\omega^{2}-\chi\right)^{2}+\psi^{2}\right)+\omega^{2}\left(\left(v^{2}-\chi\right)^{2}+\psi^{2}\right)}\right)}{4\omega\sqrt{\left(\omega^{2}-\chi\right)^{2}+\psi^{2}}}-\frac{f\left(\frac{2w_{1}\omega\sqrt{\left(\left(w_{1}^{2}-\chi\right)^{2}+\psi^{2}\right)\left(\left(\omega^{2}-\chi\right)^{2}+\psi^{2}\right)}}{w_{1}^{2}\left(\left(\omega^{2}-\chi\right)^{2}+\psi^{2}\right)+\omega^{2}\left(\left(w_{1}^{2}-\chi\right)^{2}+\psi^{2}\right)}\right)}{4\omega\sqrt{\left(\omega^{2}-\chi\right)^{2}+\psi^{2}}}
12. \int_{u}^{+\infty}\frac{1}{(t^{2}-\omega^{2})\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt+\int_{v}^{+\infty}\frac{1}{(t^{2}-\omega^{2})\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt
13. \int_{w_{1}}^{+\infty}\frac{1}{(t^{2}-\omega^{2})\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}dt+g\left(x\right)
14. \omega=0.8
15. w_{1}=\frac{v\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}-u\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}}{u^{2}-v^{2}}
16. f\left(x\right)=\operatorname{arctanh}\left(x^{\operatorname{sgn}\left(1-\left|x\right|\right)}\right)

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青青子衿

\begin{align*}
&\int_{u}^{+\infty}\frac{\mathrm{d}t}{\left(t^{2}-\omega^{2}\right)\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}

+\int_{v}^{+\infty}\frac{\mathrm{d}t}{\left(t^{2}-\omega^{2}\right)\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}\\

&\qquad=\int_{w}^{+\infty}\frac{\mathrm{d}t}{\left(t^{2}-\omega^{2}\right)\sqrt{(t^{2}-\chi)^{2}+\psi^{2}}}\\

&\qquad\qquad+\tfrac{1}{4\omega\sqrt{(\omega^{2}-\chi)^{2}+\psi^{2}}}\operatorname{arctanh}\left(\tfrac{2u\omega\sqrt{((\omega^{2}-\chi)^{2}+\psi^{2})((u^{2}-\chi)^{2}+\psi^{2})}}{u^{2}\left((\omega^{2}-\chi)^{2}+\psi^{2}\right)+\omega^{2}\left((u^{2}-\chi)^{2}+\psi^{2}\right)}\right)\\

&\qquad\qquad\qquad+\tfrac{1}{4\omega\sqrt{(\omega^{2}-\chi)^{2}+\psi^{2}}}\operatorname{arctanh}\left(\tfrac{2v\omega\sqrt{((\omega^{2}-\chi)^{2}+\psi^{2})((v^{2}-\chi)^{2}+\psi^{2})}}{v^{2}\left((\omega^{2}-\chi)^{2}+\psi^{2}\right)+\omega^{2}\left((v^{2}-\chi)^{2}+\psi^{2}\right)}\right)\\

&\qquad\qquad-\tfrac{1}{4\omega\sqrt{(\omega^{2}-\chi)^{2}+\psi^{2}}}\operatorname{arctanh}\left(\tfrac{2w\omega\sqrt{((\omega^{2}-\chi)^{2}+\psi^{2})((w^{2}-\chi)^{2}+\psi^{2})}}{w^{2}\left((\omega^{2}-\chi)^{2}+\psi^{2}\right)+\omega^{2}\left((w^{2}-\chi)^{2}+\psi^{2}\right)}\right)\\

&-\tfrac{1}{2\omega\sqrt{(\omega^{2}-\chi)^{2}+\psi^{2}}}\operatorname{arctanh}\left(\tfrac{\frac{4A(A^{2}-\omega^{4})}{\omega}\sqrt{(\omega^{2}-\chi)^{2}+\psi^{2}}uvw}{(A-\omega^{2})^{2}(A+u^{2})(A+v^{2})(A+w^{2})+(A+\omega^{2})^{2}(A-u^{2})(A-v^{2})(A-w^{2})}\right)\\

&\qquad\qquad=\frac{1}{2 \omega  \sqrt{(\omega ^2-\chi )^2+\psi ^2}}
\ln \left(\tfrac{
\begin{vmatrix}
1 & u^2 & u \sqrt{(u^2-\chi)^2+\psi ^2} \\
1 & v^2 & v \sqrt{(v^2-\chi )^2+\psi ^2} \\
1 & \omega ^2 & -\omega  \sqrt{(\omega ^2-\chi)^2+\psi ^2} \\
\end{vmatrix}
}{
\begin{vmatrix}
1 & u^2 & u \sqrt{(u^2-\chi)^2+\psi ^2} \\
1 & v^2 & v \sqrt{(v^2-\chi )^2+\psi ^2} \\
1 & \omega ^2 & \omega  \sqrt{(\omega ^2-\chi)^2+\psi ^2} \\
\end{vmatrix}
}\right)\\


\\
&\qquad\qquad\left\{\begin{split}
w&=\frac{v\sqrt{(u^{2}-\chi)^{2}+\psi^{2}}-u\sqrt{(v^{2}-\chi)^{2}+\psi^{2}}}{u^{2}-v^{2}}\\
A&=\sqrt{\chi^{2}+\psi^{2}}\\
\operatorname{arctanh}(x)&=\operatorname{arctanh}\left(x^{\operatorname{sgn}(1-\left|x\right|)}\right)
\end{split}\right.


\end{align*}

1. lis3 = {u -> 9, v -> 7, \[Chi] -> 3, \[Psi] -> 2, \[Omega] -> 1};
2. N[NIntegrate[
3.    1/((t^2 - \[Omega]^2) Sqrt[(t^2 - \[Chi])^2 + \[Psi]^2]) /.
4.     lis3, {t, u /. lis3, +Infinity}, WorkingPrecision -> 50]
5.   + NIntegrate[
6.    1/((t^2 - \[Omega]^2) Sqrt[(t^2 - \[Chi])^2 + \[Psi]^2]) /.
7.     lis3, {t, v /. lis3, +Infinity}, WorkingPrecision -> 50]
8.   - NIntegrate[
9.    1/((t^2 - \[Omega]^2) Sqrt[(t^2 - \[Chi])^2 + \[Psi]^2]) /.
10.     lis3, {t, (
11.      v Sqrt[(u^2 - \[Chi])^2 + \[Psi]^2] -
12.       u Sqrt[(v^2 - \[Chi])^2 + \[Psi]^2])/(u^2 - v^2) /.
13.      lis3, +Infinity}, WorkingPrecision -> 50], 40]
16. N[1/(2 \[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \[Psi]^2]) Log[Det[( {
17.        {1, u^2, u Sqrt[(u^2 - \[Chi])^2 + \[Psi]^2]},
18.        {1, v^2, v Sqrt[(v^2 - \[Chi])^2 + \[Psi]^2]},
19.        {1, \[Omega]^2, -\[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \
20. \[Psi]^2]}
21.       } )]/Det[( {
22.        {1, u^2, u Sqrt[(u^2 - \[Chi])^2 + \[Psi]^2]},
23.        {1, v^2, v Sqrt[(v^2 - \[Chi])^2 + \[Psi]^2]},
24.        {1, \[Omega]^2, \[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \
25. \[Psi]^2]}
26.       } )]] /. lis3, 40]
28. AarcTanh[x_] := ArcTanh[x^Sign[1 - Abs[x]]]
29. N[(1/(4 \[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \[Psi]^2])
30.          AarcTanh[(
31.          2 u*\[Omega] Sqrt[((\[Omega]^2 - \[Chi])^2 + \[Psi]^2) ((u^2 \
32. - \[Chi])^2 + \[Psi]^2)])/(
33.          u^2 ((\[Omega]^2 - \[Chi])^2 + \[Psi]^2) + \[Omega]^2 ((u^2 \
34. - \[Chi])^2 + \[Psi]^2))]
35.        +
36.        1/(4 \[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \[Psi]^2])
37.          AarcTanh[(
38.          2 v*\[Omega] Sqrt[((\[Omega]^2 - \[Chi])^2 + \[Psi]^2) ((v^2 \
39. - \[Chi])^2 + \[Psi]^2)])/(
40.          v^2 ((\[Omega]^2 - \[Chi])^2 + \[Psi]^2) + \[Omega]^2 ((v^2 \
41. - \[Chi])^2 + \[Psi]^2))]
42.        -
43.        1/(4 \[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \[Psi]^2])
44.          AarcTanh[(
45.          2 w*\[Omega] Sqrt[((\[Omega]^2 - \[Chi])^2 + \[Psi]^2) ((w^2 \
46. - \[Chi])^2 + \[Psi]^2)])/(
47.          w^2 ((\[Omega]^2 - \[Chi])^2 + \[Psi]^2) + \[Omega]^2 ((w^2 \
48. - \[Chi])^2 + \[Psi]^2))]
49.        -
50.        1/(2 \[Omega] Sqrt[(\[Omega]^2 - \[Chi])^2 + \[Psi]^2])
51.          AarcTanh[((
52.           4 A (A^2 - \[Omega]^4))/\[Omega] Sqrt[(\[Omega]^2 - \
53. \[Chi])^2 + \[Psi]^2] u*v*
54.           w)/((A - \[Omega]^2)^2 (A + u^2) (A + v^2) (A +
55.              w^2) + (A + \[Omega]^2)^2 (A - u^2) (A - v^2) (A -
56.              w^2))]) /. {
57.       W -> -((u*V - v*U )/(u - v)) - (U - V )/(u - v) w} /. {
58.      w -> (
59.       v Sqrt[(u^2 - \[Chi])^2 + \[Psi]^2] -
60.        u Sqrt[(v^2 - \[Chi])^2 + \[Psi]^2])/(u^2 - v^2)} /. {
61.     U -> Sqrt[(u^2 - \[Chi])^2 + \[Psi]^2],
62.     V -> Sqrt[(v^2 - \[Chi])^2 + \[Psi]^2],
63.     A -> Sqrt[\[Chi]^2 + \[Psi]^2]} /. lis3, 40]

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1. points = {{a, 0}, {p, P}, {q, Q}};
2. poly = InterpolatingPolynomial[points,
3.     x] - (x - a) ((
4.       p^2 Q - q^2 P - a (p*Q - q*P))/((a - p) (a - q) (p - q)) - (
5.        a (P - Q) + (p*Q - q*P))/((a - p) (a - q) (p - q)) x) //
6.   Factor
7. points = {{u, U}, {v, V}};
8. poly = InterpolatingPolynomial[points,
9.     x] - ((u*V - v*U )/(u - v) + (U - V )/(u - v) x) // Factor

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青青子衿

Pseudo-elliptic integrals, units, and torsion
Francesco Pappalardi, Alfred J. van der Poorten
arxiv.org/pdf/math/0403228

1. N[NIntegrate[1/(
2.    2 (t - 1) Sqrt[t ((t - 3)^2 + 2^2)]), {t, 5^2, +Infinity},
3.    WorkingPrecision -> 50]
4.   + NIntegrate[1/(
5.    2 (t - 1) Sqrt[t ((t - 3)^2 + 2^2)]), {t, 7^2, +Infinity},
6.    WorkingPrecision -> 50]
7.   - NIntegrate[1/(
8.    2 (t - 1) Sqrt[
9.     t ((t - 3)^2 + 2^2)]), {t, (1/
10.       24 (10 Sqrt[530] - 14 Sqrt[122]))^2, +Infinity},
11.    WorkingPrecision -> 50], 40]
13. N[1/(4 Sqrt[2])
14.    Log[((-1721 + 5 Sqrt[61]) (-3443 + 7 Sqrt[265]))/5920632], 40]
15. N[1/(2 Sqrt[a ((a - 3)^2 + 2^2)]) Log[Det[( {
16.        {1, u, Sqrt[u ((u - 3)^2 + 2^2)]},
17.        {1, v, Sqrt[v ((v - 3)^2 + 2^2)]},
18.        {1, a, -Sqrt[a ((a - 3)^2 + 2^2)]}
19.       } )]/Det[( {
20.        {1, u, Sqrt[u ((u - 3)^2 + 2^2)]},
21.        {1, v, Sqrt[v ((v - 3)^2 + 2^2)]},
22.        {1, a, Sqrt[a ((a - 3)^2 + 2^2)]}
23.       } )]] /. {u -> 5^2, v -> 7^2, a -> 1}, 40]

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\begin{gather*}

{\large\int} \frac{s_{3}(x^3+W_{3})+s_{2}(x^2+W_{2})+s_{1}(x+W_{1})}{(x^3+ux^2+vx+w)\sqrt{x^3+Ax+B}}\mathrm{d}x\\
\\
\left\{
\begin{split}
A&=u\mu\\
B&=\frac{u(u^2+18u\mu-27\mu ^2)}{108}\\
v&= -\frac{u (u+6 \mu)}{3}\\
w&=\frac{u(u^2+18u\mu+54 \mu ^2)}{27}\\
W_{1}&=\frac{u}{3}\\
W_{2}&=-\frac{u^2}{9}\\
W_{3}&=-\frac{u(2 u^2+18 u\mu+27 \mu ^2)}{27}\\
\end{split}\right.\\

\end{gather*}






\begin{align*}
x&=\frac{-36bt-144ab+(t-4a)\sqrt{t^{3}-64a^{3}-432b^{2}}}{6(t^{2}+4at+16a^{2})}\\
y&=\frac{36bt+(t+8a)\sqrt{t^{3}-64a^{3}-432b^{2}}}{6(t^{2}+4at+16a^{2})}\\
\mathrm{d}x&=\left(\frac{6bt(8a+t)}{(t^{2}+4at+16a^{2})^{2}}-\frac{(t^{2}-8at-32a^{2})\sqrt{t^{3}-64a^{3}-432b^{2}}}{6(t^{2}+4at+16a^{2})^{2}}\right.\\
&\qquad\qquad\left.+\frac{t^{2}(t-4a)}{4(t^{2}+4at+16a^{2})\sqrt{t^{3}-64a^{3}-432b^{2}}}\right)\mathrm{d}t\\
\dfrac{\mathrm{d}x}{y}&=\left(\frac{t+8a}{2(t^{2}+4at+16a^{2})}+\frac{18bt}{(t^{2}+4at+16a^{2})\sqrt{t^{3}-64a^{3}-432b^{2}}}\right)\mathrm{d}t\\

&=\left(\frac{t+8a}{2(t^{2}+4at+16a^{2})}\right.\\
&\qquad+\left(\tfrac{-1-i\sqrt{3}}{2}\right)\left(\tfrac{1}{\frac{-1+i\sqrt{3}}{2}t-4a}+c_0\right)\frac{\frac{-1+i\sqrt{3}}{2}}{\sqrt{\left({\scriptsize\frac{-1+i\sqrt{3}}{2}}t\right)^{3}-64a^{3}-432b^{2}}}\\
&\qquad\quad\left.-\left(\tfrac{-1+i\sqrt{3}}{2}\right)\left(\tfrac{1}{\frac{-1-i\sqrt{3}}{2}t-4a}+c_0\right)\frac{\frac{-1-i\sqrt{3}}{2}}{\sqrt{\left({\scriptsize\frac{-1-i\sqrt{3}}{2}}t\right)^{3}-64a^{3}-432b^{2}}}\right)\mathrm{d}t\\

\end{align*}

1. y^3 - (x^3 + a*x + b) /. {
2.    x -> (-36 b*t - 144 a*b + (t - 4 a) s)/(6 (t^2 + 4 a*t + 16 a^2)),
3.    y -> (36 b*t + (t + 8 a) s)/(6 (t^2 + 4 a*t + 16 a^2))
4.    } // Factor
5. D[(-36 b*t - 144 a*b + (t - 4 a) Sqrt[t^3 - 64 a^3 - 432 b^2])/(
6.    6 (t^2 + 4 a*t + 16 a^2)), t] - (
7.    
8.    (6 b t (8 a + t))/(16 a^2 + 4 a t +
9.       t^2)^2 + ((32 a^2 + 8 a t - t^2) Sqrt[t^3 - 64 a^3 - 432 b^2])/(
10.     6 (16 a^2 + 4 a t + t^2)^2) + ((t - 4 a) (3 t^2)/(
11.      2 Sqrt[t^3 - 64 a^3 - 432 b^2]))/(
12.     6 (16 a^2 + 4 a t + t^2))) // Factor
14. ((6 b t (8 a + t))/(16 a^2 + 4 a t +
15.      t^2)^2 + ((32 a^2 + 8 a t - t^2) Sqrt[t^3 - 64 a^3 - 432 b^2])/(
16.    6 (16 a^2 + 4 a t + t^2)^2) + ((t - 4 a) (3 t^2)/(
17.     2 Sqrt[t^3 - 64 a^3 - 432 b^2]))/(6 (16 a^2 + 4 a t + t^2)))/((
18.   36 b*t + (t + 8 a) Sqrt[t^3 - 64 a^3 - 432 b^2])/(
19.   6 (t^2 + 4 a*t + 16 a^2))) /. {a -> 1, b -> -1, t -> 10} // N
20. (36 b*t + (t + 8 a)*s)/(2 (16 a^2 + 4 a t + t^2) s) /.
21.    s -> Sqrt[t^3 - 64 a^3 - 432 b^2] /. {a -> 1, b -> -1, t -> 10} // N
22. (18 b*t)/( (t^2 + 4*a*t + 16 a^2) Sqrt[t^3 - 64 a^3 - 432 b^2])
24. 6 I Sqrt[3]
25.    b ((-1 - I Sqrt[3])/
26.      2 (1/((-1 + I Sqrt[3])/2 t - 4 a) + c) ((-1 + I Sqrt[3])/2)/
27.      Sqrt[((-1 + I Sqrt[3])/2 t)^3 - 64 a^3 - 432 b^2]
28.     - (-1 + I Sqrt[3])/
29.      2 (1/((-1 - I Sqrt[3])/2 t - 4 a) + c) ((-1 - I Sqrt[3])/2)/
30.      Sqrt[((-1 - I Sqrt[3])/2 t)^3 - 64 a^3 -
31.       432 b^2]) // FullSimplify

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