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Last edited by 青青子衿 at 2025-3-28 10:57:22kuing.cjhb.site/forum.php?mod=redirect&go … =11209&pid=61234
\begin{align*}
\wp ^{-1}(z;g_2,g_3)&=\int_{z}^{\infty}\frac{\mathrm{d}t}{\sqrt{4t^3-g_2t-g_3}}\\
\zeta(\wp ^{-1}(z;g_2,g_3);g_2,g_3)&=\frac{1}{\wp ^{-1}(z;g_2,g_3)}-\int_0^{z}\frac{u-\frac{1}{[\wp^{-1}\left(u;g_2,g_3\right)]^2}}{\sqrt{4u^3-g_2u-g_3}}\mathrm{d}u\\
\Xi(z,\xi;g_2,g_3)&=\frac{1}{\sqrt{4\xi^3-g_2\xi-g_3}}\ln\bigg(\frac{\sigma (\wp^{-1}\left(z;g_2,g_3\right)-\wp^{-1}\left(\xi;g_2,g_3\right) ;g_2,g_3)}{\sigma (\wp^{-1}\left(z;g_2,g_3\right)+\wp^{-1}\left(\xi;g_2,g_3\right) ;g_2,g_3)}\bigg)\\
&\qquad+\frac{2}{\sqrt{4\xi^3-g_2\xi-g_3}}\wp ^{-1}(z;g_2,g_3)\cdot\zeta(\wp ^{-1}(z;g_2,g_3);g_2,g_3)\\
&=\int_0^{z}\frac{\mathrm{d}u}{(u-\xi)\sqrt{4u^3-g_2u-g_3}}\\
\end{align*}
\begin{align*}
\zeta \left(z;g_2,g_3\right)
&=\frac{1}{z}-\int_0^z \left(\wp \left(t;g_2,g_3\right)-\frac{1}{t^2}\right) \mathrm{d}t\\
&=\frac{1}{z}-\int_0^z \frac{\wp \left(t;g_2,g_3\right)-\frac{1}{t^2}}{\wp'\!\left(t;g_2,g_3\right)} \mathrm{d}(\wp\left(t;g_2,g_3\right))\\
&=\frac{1}{z}-\int_0^{\wp\left(z;g_2,g_3\right)}\frac{u-\frac{1}{[\wp^{-1}\left(u;g_2,g_3\right)]^2}}{\sqrt{4u^3-g_2u-g_3}}\mathrm{d}u\\
\end{align*}
\begin{align*}
\frac{\wp '(\xi ;g_2,g_3)}{\wp (z;g_2,g_3)-\wp (\xi ;g_2,g_3)}&=\frac{\partial}{\partial{z}}\left[\ln\left(\frac{\sigma (z-\xi ;g_2,g_3)}{\sigma (z+\xi ;g_2,g_3)}\right)+2 z \zeta (\xi ;g_2,g_3)\right]\\
\frac{{\sqrt{4\xi^3-g_2\xi-g_3}}}{(z-\xi){\sqrt{4z^3-g_2z-g_3}}}&=\frac{\partial}{\partial{z}}\Bigg[
\ln\bigg(\frac{\sigma (\wp^{-1}\left(z;g_2,g_3\right)-\wp^{-1}\left(\xi;g_2,g_3\right) ;g_2,g_3)}{\sigma (\wp^{-1}\left(z;g_2,g_3\right)+\wp^{-1}\left(\xi;g_2,g_3\right) ;g_2,g_3)}\bigg)
\\
&\qquad\qquad
+2\wp^{-1}\left(z;g_2,g_3\right)\cdot\zeta (\wp^{-1}\left(\xi;g_2,g_3\right) ;g_2,g_3)
\Bigg]\\
\end{align*}
Weierstrassche Normalintegrale
Wolfgang Gröbner, - Integraltafel_ Erster Teil Unbestimmte Integrale (1975).pdf
Eugene & Emde, Fritz Jahnke - Tables of Functions with Formulae and Curves (1945).pdf
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