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ellipticK を含む積分

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$$\newcommand{BA}[0]{\begin{align*}} \newcommand{BE}[0]{\begin{equation}} \newcommand{bl}[0]{\boldsymbol} \newcommand{BM}[0]{\begin{matrix}} \newcommand{D}[0]{\displaystyle} \newcommand{EA}[0]{\end{align*}} \newcommand{EE}[0]{\end{equation}} \newcommand{EM}[0]{\end{matrix}} \newcommand{h}[0]{\boldsymbol{h}} \newcommand{k}[0]{\boldsymbol{k}} \newcommand{L}[0]{\left} \newcommand{l}[0]{\boldsymbol{l}} \newcommand{m}[0]{\boldsymbol{m}} \newcommand{n}[0]{\boldsymbol{n}} \newcommand{R}[0]{\right} \newcommand{vep}[0]{\varepsilon} $$

命題

$\BA\D \D \int_0^1 K(x)^2\,dx=\frac{1}{2}\int_0^1 K'(x)^2\,dx=\frac{\pi^4}{32}\sum_{n=0}^\infty (4n+1)\frac{\binom{2n}{n}^6}{2^{12n}} \EA$

命題

$\BA\D \int_0^1 \frac{K(x)}{\sqrt{1-x^2}}\ln\frac{1}{1-x^2}\,dx=\frac{\Gamma(\frac{1}{4})^4}{24} \EA$

命題

$\BA\D \int_0^1 \frac{xK(x)^2}{\sqrt{1-x^2}}\ln\frac{1}{1-x^2}\,dx =\frac{\pi^4}{4}\sum_{n=0}^\infty \frac{\binom{2n}{n}^4}{2^{8n}} \EA$

命題

$\BA\D \int_0^1 \frac{K(x)\ln\frac{1}{x}}{1-x^2}\,dx=\frac{7}{4}\zeta(3) \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)\ln\frac{1}{x}}{1-x^2}\,dx=\frac{\pi^3}{8} \EA$

命題

$\BA\D \int_0^1 \frac{xK(x)K'(x)}{1-t^2x^2}\,dx=\frac{\pi}{4}K(t)^2 \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)}{1-t^2x^2}\,dx=\frac{\pi}{2}K(t) \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)\tanh^{-1}x}{x}\,dx=\pi\beta(2) \EA$

命題

$\BA\D \int_0^1 \frac{K(x)K'(x)}{x}\ln\frac{1}{1-x^2}\,dx=\frac{7}{8}\pi\zeta(3) \EA$

命題

$\BA\D \int_0^1 \frac{K(x)}{1+x}\,dx=\frac{\pi^2}{8} \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)}{1+x}\,dx=2\beta(2) \EA$

命題

$\BA\D \int_0^1 K'(x)^3\,dx=\frac{10}{3}\int_0^1 K(x)^3\,dx=\frac{5}{2}\int_0^1 K(\sqrt{x})^3\,dx=6\int_0^1 xK(x)^2K'(x)\,dx=3\int_0^1 K(x)^2K'(x)\,dx=2\int_0^1 K(x)K'(x)^2\,dx=\frac{{\Gamma(\frac{1}{4})}^8}{128\pi^2} \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)^2}{\sqrt{1-x^2}}\,dx=\int_0^1 \frac{K(x)^2}{\sqrt{1-x^2}}\,dx=2\int_0^1 K(x)K'(x)\,dx=\frac{\pi^3}{4}\sum_{n=0}^\infty \frac{\binom{2n}{n}^4}{2^{8n}} \EA$

命題

$\BA\D \frac{93}{8}\zeta(5)=\int_0^1 (2x-1)K(\sqrt{x})^4\,dx \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)}{\sqrt{1+x}}\,dx =\frac{{\Gamma(\frac{1}{8})}^2{\Gamma(\frac{3}{8})}^2}{32\pi}-\frac{1}{2}\sum_{n=0}^\infty \frac{(4n+1)\binom{4n}{2n}}{(2n+1)^3\binom{2n}{n}^2} \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)}{\sqrt{1-x}}\,dx =\frac{{\Gamma(\frac{1}{8})}^2{\Gamma(\frac{3}{8})}^2}{32\pi}+\frac{1}{2}\sum_{n=0}^\infty \frac{(4n+1)\binom{4n}{2n}}{(2n+1)^3\binom{2n}{n}^2}\EA$

命題

$\BA\D \int_0^1 \frac{K(x)}{\sqrt{1+x}}\,dx=\frac{1}{3}\int_0^1 \frac{K(x)}{\sqrt{1-x}}\,dx=\frac{{\Gamma(\frac{1}{8})}^2{\Gamma(\frac{3}{8})}^2}{48\sqrt{2}\,\pi} \EA$

命題

$\BA\D \int_0^1 (2x-1)K(\sqrt{x})^3K'(\sqrt{x})\,dx=\frac{\pi^5}{64} \EA$

命題

$\BA\D \int_0^1 (2x-1)K(\sqrt{x})^2K'(\sqrt{x})\,dx=\frac{32\,\pi^6}{\Gamma\L(\frac{1}{4}\R)^8} \EA$

命題

$\BA\D \int_0^1 \frac{1-x}{1+x}K(x)^2\,dx=\frac{7}{8}\zeta(3) \EA$

命題

$\BA\D \int_0^1 K(x)^2\,dx=\int_0^1 \frac{4K(x)^2-K'(x)^2}{1+x}\,dx \EA$

命題

$\BA\D \int_0^1 \frac{K(x)\ln\frac{1}{x}}{1+x}\,dx=\frac{7}{2}\zeta(3)-\pi\beta(2) \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)^2}{1+x^2}\,dx-\int_0^1 K(x)^2\,dx=\int_0^1 \frac{K(\sqrt{x})K'(\sqrt{x})}{1+x}\,dx=\frac{\Gamma\left(\frac{1}{4}\right)^4}{64} \EA$

命題

$\BA\D \int_0^1 \frac{6K(x)^2K'(x)+2K(x)K'(x)^2-K'(x)^3}{1+x^2}\,dx=\frac{3}{256}\frac{\Gamma\left(\frac{1}{4}\right)^8}{\pi^2} \EA$

命題

$\BA\D \int_0^1 \frac{K(x)\ln^2\frac{1}{x}}{1-x^2}\,dx=4\beta(2)^2 \EA$

命題

$\BA\D \int_0^1 \frac{K'(x)\ln^2\frac{1}{x}}{1-x^2}\,dx=\pi^2\beta(2) \EA$

命題

$\BA\D 8\int_0^1 \frac{K(x)^2K'(x)^2}{1+x^2}\,dx=\int_0^1 \L(2K(x)^2K'(x)^2+K'(x)^4\R)\,dx \EA$

命題

$\BA\D \int_0^1 \frac{24K(x)^4+6K'(x)^4}{1+x^2}\,dx=\int_0^1 \L(8K(x)^4+7K'(x)^4\R)\,dx \EA$

命題

$\BA\D \int_0^1 \frac{2865K'(x)^5+31608K(x)K'(x)^4-28246K(x)^5}{1+x}\,dx=\frac{753}{128}\pi^6 \EA$

命題

$\BA\D \int_0^1 \frac{xK(x)^2}{\sqrt[4]{1-x^2}}\,dx=4\int_0^1 \left(K(x)K'(x)-K(x)^2\right)\,dx \EA$

命題

$\BA\D \int_0^1 \frac{xK'(x)^2}{\sqrt[4]{1-x^2}}\,dx=\int_0^1 \left(K(x)K'(x)-\frac{1}{2}K(x)^2\right)\,dx \EA$

命題

$\BA\D \int_0^1 \frac{xK(x)K'(x)}{\sqrt[4]{1-x^2}}\,dx=\int_0^1 \left(2K(x)^2-K(x)K'(x)\right)\,dx \EA$

投稿日:2023929

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