双曲无穷级数与雅可比西塔函数

青青子衿

\begin{align*}
\sum_{n=0}^{+\infty}\frac{1}{\cosh\left(n\pi\rho\right)}&=\dfrac{\vartheta_{3}^2(0,e^{-\pi\rho})}{2}+\dfrac{1}{2}\\
\sum_{n=0}^{+\infty}\frac{1}{\cosh^{2}\!\left(n\pi\rho\right)}
&=-\frac{\left.\partial_z^2\vartheta_{2}\left(z,e^{-\pi\rho}\right)\right|_{z=0}}{2\vartheta_{2}\left(0,e^{-\pi\rho}\right)}+\dfrac{1}{2}\\
\sum_{n=0}^{+\infty}\frac{1}{\cosh^{3}\!\left(n\pi\rho\right)}&=\frac{\vartheta _3^2(0,e^{-\pi\rho})\vartheta _4^4(0,e^{-\pi\rho})}{4}+\frac{\vartheta _3^2(0,e^{-\pi\rho})}{4}+\frac{1}{2}\\
\sum_{n=0}^{+\infty}\frac{1}{\cosh^{4}\!\left(n\pi\rho\right)}&=\frac{\vartheta_3^4(0,e^{-\pi\rho})\vartheta_4^4(0,e^{-\pi\rho})}{6}-\frac{\left.\partial_z^2\vartheta_2(z,e^{-\pi\rho})\right|_{z=0}}{3 \vartheta _2(0,e^{-\pi\rho})}+\frac{1}{2}\\
\sum_{n=0}^{+\infty}\frac{1}{\cosh^{5}\!\left(n\pi\rho\right)}&=\frac{\vartheta _3^6\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{12}+\frac{\vartheta _3^2\left(0,e^{-\pi  \rho }\right) \vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{48}\\
&\qquad\quad+\frac{5\vartheta _3^2\left(0,e^{-\pi\rho}\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{24}+\frac{3\vartheta _3^2\left(0,e^{-\pi  \rho }\right)}{16}+\frac{1}{2}\\
\sum_{n=0}^{+\infty}\frac{1}{\cosh^{6}\!\left(n\pi\rho\right)}
&=\frac{\vartheta _3^8\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{30}+\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{30}\\
&\qquad\quad\>\>+\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{6}+\frac{1}{2}\\
&\qquad\qquad\quad-\frac{4\left.\partial_z^2\vartheta_{2}\left(z,e^{-\pi\rho}\right)\right|_{z=0}}{15\vartheta_{2}\left(0,e^{-\pi\rho}\right)}\\

\end{align*}

1. R = EllipticTheta[3, 0, Exp[-Pi*\[Rho]]];
2. S = EllipticTheta[4, 0, Exp[-Pi*\[Rho]]];
3. T = D[EllipticTheta[2, z, Exp[-Pi*\[Rho]]], z, z]/
4.     EllipticTheta[2, 0, Exp[-Pi*\[Rho]]] /. z -> 0;
5. NSum[1/Cosh[n*Pi*\[Rho]] /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
6. WorkingPrecision -> 15]
7. NumberForm[R^2/2 + 1/2 /. {\[Rho] -> 0.23}, 15]
8. NSum[1/Cosh[n*Pi*\[Rho]]^2 /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
9. WorkingPrecision -> 15]
10. NumberForm[-T/2 + 1/2 /. {\[Rho] -> 0.23}, 15]
11. NSum[1/Cosh[n*Pi*\[Rho]]^3 /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
12. WorkingPrecision -> 15]
13. NumberForm[R^2 S^4 /4+R^2 /4 + 1/2 /. {\[Rho] -> 0.23}, 15]
14. NSum[1/Cosh[n*Pi*\[Rho]]^4 /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
15. WorkingPrecision -> 15]
16. NumberForm[R^4 S^4/6 - T/3 + 1/2 /. {\[Rho] -> 0.23}, 15]
17. NSum[1/Cosh[n*Pi*\[Rho]]^5 /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
18. WorkingPrecision -> 15]
19. NumberForm[
20. R^6 S^4/12 + R^2 S^8/48 + 5 R^2 S^4/24 + 3 R^2/16 +
21.    1/2 /. {\[Rho] -> 0.23}, 15]
22. NSum[1/Cosh[n*Pi*\[Rho]]^6 /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
23. WorkingPrecision -> 15]
24. NumberForm[
25. R^8 S^4/30 + R^4 S^8/30 + R^4 S^4/6 - 4 T/15 +
26.    1/2 /. {\[Rho] -> 0.23}, 15]

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www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html#SeriesxofxHyperbolicxFunctions

青青子衿

\begin{align*}
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh\left(n\pi\rho\right)}&=\dfrac{\vartheta_{4}^2(0,e^{-\pi\rho})}{2}+\dfrac{1}{2}\\
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh ^2(n\pi\rho)}&=\dfrac{\vartheta_{3}^2(0,e^{-\pi\rho})\vartheta_{4}^2(0,e^{-\pi\rho})}{2}+\dfrac{1}{2}\\
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh ^3(n\pi\rho)}&=\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{4}
+\frac{\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{4}+\frac{1}{2}\\
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh^4\left(n\pi\rho\right)}
&=\frac{\vartheta _3^6\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{12}+\frac{\vartheta _3^2\left(0,e^{-\pi  \rho }\right)\vartheta _4^6\left(0,e^{-\pi  \rho }\right)}{12}\\
&\qquad\quad+\frac{\vartheta _3^2\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{3}+\frac{1}{2}\\
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh^5\left(n\pi\rho\right)}
&=\frac{\vartheta _3^8\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{48}+\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^6\left(0,e^{-\pi  \rho }\right)}{12}\\
&\qquad+\frac{5\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right) }{24}+\frac{3\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{16}+\frac{1}{2}\\
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh^6\left(n\pi\rho\right)}
&=\frac{\vartheta _3^{10}\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right) }{240}+\frac{7\vartheta _3^6\left(0,e^{-\pi  \rho }\right)\vartheta _4^6\left(0,e^{-\pi  \rho }\right)}{120} \\
&\qquad+\frac{\vartheta _3^6\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{12}
+\frac{\vartheta _3^2\left(0,e^{-\pi  \rho }\right)\vartheta _4^{10}\left(0,e^{-\pi  \rho }\right)}{240}\\
&\qquad\qquad+\frac{\vartheta _3^2\left(0,e^{-\pi  \rho }\right)\vartheta _4^6\left(0,e^{-\pi  \rho }\right)}{12}
+\frac{1}{2}\\
&\qquad\qquad\qquad+\frac{4\vartheta _3^2\left(0,e^{-\pi  \rho }\right)\vartheta _4^2\left(0,e^{-\pi  \rho }\right)}{15}\\
\end{align*}

1. R = EllipticTheta[3, 0, Exp[-Pi*\[Rho]]];
2. S = EllipticTheta[4, 0, Exp[-Pi*\[Rho]]];
3. T = D[EllipticTheta[2, z, Exp[-Pi*\[Rho]]], z, z]/
4.     EllipticTheta[2, 0, Exp[-Pi*\[Rho]]] /. z -> 0;
5. NSum[(-1)^n/Cosh[n*Pi*\[Rho]] /. \[Rho] -> 0.23, {n, 0, +\[Infinity]},
6.   WorkingPrecision -> 15]
7. NumberForm[S^2/2 + 1/2 /. {\[Rho] -> 0.23}, 15]
8. NSum[(-1)^n/Cosh[n*Pi*\[Rho]]^2 /. \[Rho] -> 0.23, {n,
9.   0, +\[Infinity]}, WorkingPrecision -> 15]
10. NumberForm[R^2 S^2/2 + 1/2 /. {\[Rho] -> 0.23}, 15]
11. NSum[(-1)^n/Cosh[n*Pi*\[Rho]]^3 /. \[Rho] -> 0.23, {n,
12.   0, +\[Infinity]}, WorkingPrecision -> 15]
13. NumberForm[R^4 S^2/4 + S^2/4 + 1/2 /. {\[Rho] -> 0.23}, 15]
14. NSum[(-1)^n/Cosh[n*Pi*\[Rho]]^4 /. \[Rho] -> 0.23, {n,
15.   0, +\[Infinity]}, WorkingPrecision -> 15]
16. NumberForm[
17. R^6 S^2/12 + R^2 S^6/12 + R^2 S^2/3 + 1/2 /. {\[Rho] -> 0.23}, 15]
18. NSum[(-1)^n/Cosh[n*Pi*\[Rho]]^5 /. \[Rho] -> 0.23, {n,
19.   0, +\[Infinity]}, WorkingPrecision -> 15]
20. NumberForm[
21. R^8 S^2/48 + R^4 S^6/12 + 5 R^4 S^2/24 + 3 S^2/16 +
22.    1/2 /. {\[Rho] -> 0.23}, 15]
23. NSum[(-1)^n/Cosh[n*Pi*\[Rho]]^6 /. \[Rho] -> 0.23, {n,
24.   0, +\[Infinity]}, WorkingPrecision -> 15]
25. NumberForm[
26. R^10 S^2/240 + 7 R^6 S^6/120 + R^6 S^2/12 + R^2 S^10/240 +
27.    R^2 S^6/12 + 4 R^2 S^2/15 + 1/2 /. {\[Rho] -> 0.23}, 15]

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青青子衿

青青子衿 发表于 2023-5-17 10:02


\begin{align*}
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh\left(n\pi\rho\right)}&=\dfrac{\vartheta_{4}^2(0,e^{-\pi\rho})}{2}+\dfrac{1}{2}\\
\sum_{n=0}^{+\infty}\dfrac{(-1)^n}{\cosh ^2(n\pi\rho)}&=\dfrac{\vartheta_{3}^2(0,e^{-\pi\rho})\vartheta_{4}^2(0,e^{-\pi\rho})}{2}+\dfrac{1}{2}\\
\end{align*}

\begin{align*}
\sum_{n=1}^{+\infty}\dfrac{1}{\sinh ^2(n\pi\rho)}&=\frac{\left.\partial_z^2\vartheta _2(z,e^{-\pi\rho})\right|_{z=0}}{2 \vartheta _2(0,e^{-\pi\rho})}+\frac{\vartheta _3^4(0,e^{-\pi\rho})}{6}+\frac{\vartheta _4^4(0,e^{-\pi\rho})}{6} +\frac{1}{6}\\
\sum_{n=1}^{+\infty}\dfrac{(-1)^{n-1}}{\sinh ^2(n\pi\rho)}&=\frac{\vartheta _3^4(0,e^{-\pi\rho})}{12}+\frac{\vartheta _4^4(0,e^{-\pi\rho})}{12} -\frac{1}{6}\\

\sum_{n=1}^{+\infty}\dfrac{1}{\sinh ^4(n\pi\rho)}&=
\frac{\vartheta _3^8\left(0,e^{-\pi  \rho }\right)}{90} -\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{90}\\
&\qquad-\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)}{9}+\frac{\vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{90}-\frac{\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{9}\\
&\qquad\qquad-\frac{\left.\partial_z^2\vartheta _2(z,e^{-\pi\rho})\right|_{z=0}}{3\vartheta _2(0,e^{-\pi\rho})}-\frac{11}{90}\\

\sum_{n=1}^{+\infty}\dfrac{(-1)^{n-1}}{\sinh ^4(n\pi\rho)}
&=\frac{7\vartheta _3^8\left(0,e^{-\pi \rho}\right)}{720} -\frac{11\vartheta _3^4\left(0,e^{-\pi \rho}\right)\vartheta _4^4\left(0,e^{-\pi \rho}\right)}{360}\\
&\qquad-\frac{\vartheta _3^4\left(0,e^{-\pi \rho}\right)}{18} +\frac{7\vartheta _4^8\left(0,e^{-\pi \rho}\right)}{720}\\
&\qquad\qquad-\frac{\vartheta _4^4\left(0,e^{-\pi \rho}\right)}{18} +\frac{11}{90}\\

\sum_{n=1}^{+\infty}\dfrac{1}{\sinh ^6(n\pi\rho)}
&=\frac{\vartheta _3^{12}\left(0,e^{-\pi  \rho }\right)}{945}
-\frac{\vartheta _3^8\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{630}\\
&\qquad-\frac{\vartheta _3^8\left(0,e^{-\pi  \rho }\right)}{90}-\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{630}\\
&\qquad\quad+\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{90}+\frac{4\vartheta _3^4\left(0,e^{-\pi  \rho }\right)}{45}\\
&\qquad\qquad+\frac{\vartheta _4^{12}\left(0,e^{-\pi  \rho }\right)}{945}
-\frac{\vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{90}
+\frac{191}{1890}\\
&\qquad\qquad\quad+\frac{4\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{45}
+\frac{4\left.\partial_z^2\vartheta _2\left(z,e^{-\pi  \rho }\right)\right|_{z=0}}{15 \vartheta _2\left(0,e^{-\pi  \rho }\right)}\\

\sum_{n=1}^{+\infty}\dfrac{(-1)^{n-1}}{\sinh ^6(n\pi\rho)}
&=\frac{31 \vartheta _3^{12}\left(0,e^{-\pi  \rho }\right)}{30240} -\frac{\vartheta _3^8\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{2016}\\
&\qquad-\frac{7\vartheta _3^8\left(0,e^{-\pi  \rho }\right)}{720}-\frac{\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{2016}\\
&\qquad\quad+\frac{11\vartheta _3^4\left(0,e^{-\pi  \rho }\right)\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{360}
+\frac{2\vartheta _3^4\left(0,e^{-\pi  \rho }\right)}{45}\\
&\qquad\qquad+\frac{31 \vartheta _4^{12}\left(0,e^{-\pi  \rho }\right)}{30240}-\frac{7\vartheta _4^8\left(0,e^{-\pi  \rho }\right)}{720}\\
&\qquad\qquad\qquad+\frac{2\vartheta _4^4\left(0,e^{-\pi  \rho }\right)}{45}-\frac{191}{1890}\\

\end{align*}

1. R = EllipticTheta[3, 0, Exp[-Pi*\[Rho]]];
2. S = EllipticTheta[4, 0, Exp[-Pi*\[Rho]]];
3. T = D[EllipticTheta[2, z, Exp[-Pi*\[Rho]]], z, z]/
4.     EllipticTheta[2, 0, Exp[-Pi*\[Rho]]] /. z -> 0;
5. NSum[1/Sinh[n*Pi*\[Rho]]^2 /. \[Rho] -> 0.23, {n, 1, +\[Infinity]},
6. WorkingPrecision -> 15]
7. NumberForm[T/2 + R^4/6 + S^4/6 + 1/6 /. {\[Rho] -> 0.23}, 15]
8. NSum[(-1)^(n - 1)/Sinh[n*Pi*\[Rho]]^2 /. \[Rho] -> 0.23, {n,
9.   1, +\[Infinity]}, WorkingPrecision -> 15]
10. NumberForm[R^4/12 + S^4/12 - 1/6 /. {\[Rho] -> 0.23}, 15]
11. NSum[1/Sinh[n*Pi*\[Rho]]^4 /. \[Rho] -> 0.23, {n, 1, +\[Infinity]},
12. WorkingPrecision -> 15]
13. NumberForm[
14. R^8/90 - R^4 S^4/90 - R^4/9 + S^8/90 - S^4/9 - T/3 -
15.    11/90 /. {\[Rho] -> 0.23}, 15]
16. NSum[(-1)^(n - 1)/Sinh[n*Pi*\[Rho]]^4 /. \[Rho] -> 0.23, {n,
17.   1, +\[Infinity]}, WorkingPrecision -> 15]
18. NumberForm[
19. 7 R^8/720 - 11 R^4 S^4/360 - R^4/18 + 7 S^8/720 - S^4/18 +
20.    11/90 /. {\[Rho] -> 0.23}, 15]
21. NSum[1/Sinh[n*Pi*\[Rho]]^6 /. \[Rho] -> 0.23, {n, 1, +\[Infinity]},
22. WorkingPrecision -> 15]
23. NumberForm[
24. R^12/945 - R^8 S^4/630 - R^8/90 - R^4 S^8/630 + R^4 S^4/90 +
25.    4 R^4/45 + S^12/945 - S^8/90 + 191/1890 + 4 S^4/45 +
26.    4 T/15 /. {\[Rho] -> 0.23}, 15]
27. NSum[(-1)^(n - 1)/Sinh[n*Pi*\[Rho]]^6 /. \[Rho] -> 0.23, {n,
28.   1, +\[Infinity]}, WorkingPrecision -> 15]
29. NumberForm[
30. 31 R^12/30240 - R^8 S^4/2016 - 7 R^8/720 - R^4 S^8/2016 +
31.    11 R^4 S^4/360 + 2 R^4/45 + 31 S^12/30240 - 7 S^8/720 - 191/1890 +
32.    2 S^4/45 /. {\[Rho] -> 0.23}, 15]

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青青子衿

青青子衿 发表于 2023-5-25 17:39
\begin{align*}
\sum_{n=1}^{+\infty}\dfrac{1}{\sinh ^2(n\pi\rho)}&=\frac{\left.\partial_z^2\vartheta _2(z,e^{-\pi\rho})\right|_{z=0}}{2 \vartheta _2(0,e^{-\pi\rho})}+\frac{\vartheta _3^4(0,e^{-\pi\rho})}{6}+\frac{\vartheta _4^4(0,e^{-\pi\rho})}{6} +\frac{1}{6}
\end{align*}

\begin{align*}
\sum_{n=1}^{+\infty}\dfrac{1}{\cosh\left(\frac{3\pi\,\!n}{2}\right)}
&=\frac{\left(1+\sqrt{2}+\sqrt{3}-\sqrt[4]{3}\right) \sqrt{2 \sqrt{3}} \>\>\Gamma^2\left(\frac{1}{4}\right)}{24\,\pi  \sqrt{\pi }}-\frac{1}{2}\\
\sum_{n=1}^{+\infty}\dfrac{1}{\cosh\left(\frac{5\pi\,\!n}{2}\right)}
&=\frac{\left(4+3 \sqrt{2}+2 \sqrt{5}-2 \sqrt{2 \sqrt{5}}\,\right) \Gamma^2\!\left(\frac{1}{4}\right)}{40\,\pi\sqrt{\pi }}-\frac{1}{2}\\
\end{align*}

\begin{align*}
\color{black}{\int_{0}^{+\infty}\frac{\cos \left(\pi\,\!x^2\right)}{\cosh (\pi\,\!x) \left[\cosh (\pi)-\cos(\pi\,\!x)\right]}\,\mathrm{d}x}&\color{black}{=}\color{black}{\frac{1}{\sqrt{2} \sinh (\pi )}\left[\frac{2+\cosh \left(\frac{\pi }{2}\right)}{\sinh \left(\frac{\pi }{2}\right)}-\frac{\left(1+\sqrt{2}\right) \Gamma^2\!\left(\frac{1}{4}\right)}{4\,\pi\sqrt{\pi }}\right]\,}\\
\color{black}{\int_{0}^{+\infty}\frac{\cos \left(\pi\,\!x^2\right)}{\cosh (\pi\,\!x) \left[\cosh (2 \pi )-\cos(2\pi\,\!x)\right]}\,\mathrm{d}x}&\color{black}{=}\color{black}{\dfrac{1}{\sqrt{2}\sinh (2\pi)}\left[\frac{\cosh \left(\frac{\pi }{2}\right)}{\sinh \left(\frac{\pi }{2}\right)}-\frac{\Gamma^2\!\left(\frac{1}{4}\right)}{4\,\pi \sqrt{\pi\,}}\right]\,}\\
\color{black}{\int_{0}^{+\infty}\frac{\cos \left(\pi\,\!x^2\right)}{\cosh (\pi\,\!x) \left[\cosh (3\pi )-\cos(3\pi\,\!x)\right]}\,\mathrm{d}x}&\color{black}{=}\color{black}{\frac{1}{\sqrt{2} \sinh (3 \pi )}\left[\frac{2+\cosh \left(\frac{3 \pi }{2}\right)}{\sinh \left(\frac{3 \pi }{2}\right)}-\frac{\left(1+2 \sqrt{2}+\sqrt{3}-2\sqrt[4]{3}\right)\sqrt{2 \sqrt{3}}\>\>\Gamma^2\!\left(\frac{1}{4}\right)}{24\,\pi\sqrt{\pi }}\right]}\\
\color{black}{\int_{0}^{+\infty}\frac{\cos \left(\pi\,\!x^2\right)}{\cosh (\pi\,\!x) \left[\cosh (4\pi )-\cos(4\pi\,\!x)\right]}\,\mathrm{d}x}&\color{black}{=}\color{black}{\frac{1}{\sqrt{2}\sinh (4\pi)}\left[\frac{\cosh (\pi )}{\sinh (\pi )}-\frac{\left(1+\sqrt{2}\right)\sqrt{2}\>\>\Gamma^2\!\left(\frac{1}{4}\right)}{16\,\pi\sqrt{\pi }}\right]}\\
\color{black}{\int_{0}^{+\infty}\frac{\cos \left(\pi\,\!x^2\right)}{\cosh (\pi\,\!x) \left[\cosh (5\pi )-\cos(5\pi\,\!x)\right]}\,\mathrm{d}x}&\color{black}{=}\color{black}{\frac{1}{\sqrt{2}\sinh (5\pi)}\left[\frac{2+\cosh \left(\frac{5\pi }{2}\right)}{\sinh \left(\frac{5\pi }{2}\right)}-\frac{\left(2+3 \sqrt{2}+\sqrt{5}-2 \sqrt{2 \sqrt{5}}\,\right)\Gamma^2\!\left(\frac{1}{4}\right)}{20\,\pi\sqrt{\pi }}\right]}
\end{align*}

青青子衿

\begin{align*}
\color{black}{\sum_{k=-\infty}^{+\infty}\frac{1}{\cosh({\small\sqrt{2}}\,k\pi)+\frac{\>\small1}{\small\sqrt{2}}}=\frac{{\small2}-\sqrt{\scriptsize2\sqrt{2}}+\sqrt{\scriptsize6+5\sqrt{2}-4\sqrt{\sqrt{2}}-4\sqrt{2\sqrt{2}}}}{\small8 \pi  \sqrt{\pi }}}\,\Gamma\left(\tfrac{1}{8}\right) \Gamma\left(\tfrac{3}{8}\right)
\end{align*}



\begin{align*}
\prod_{j=1}^{+\infty}(1-q^j(k))=\frac{k^{1/12}(1-k^{2})^{1/6}}{2^{1/6}q^{1/24}(k)}\sqrt{\frac{2K\left(k\right)}{\pi}}\\
\end{align*}

1. NProduct[
2. 1 - EllipticNomeQ[ModularLambda[I Sqrt[2]]]^k, {k, 1, +Infinity},
3. WorkingPrecision -> 60]
4. N[(ModularLambda[I Sqrt[2]]^(1/24) (1 - ModularLambda[I Sqrt[2]])^(
5.    1/6))/(2^(1/6) EllipticNomeQ[ModularLambda[I Sqrt[2]]]^(1/24))
6.    Sqrt[(2 EllipticK[ModularLambda[I Sqrt[2]]])/\[Pi]], 50]

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