타원적분의 singular value k

수학노트
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개요

  • 자연수 \(n \) 에 대하여, 다음을 만족시키는\(k_n\)를 타원적분의 singular value 라 한다\[\frac{K'}{K}(k_n):=\frac{K(\sqrt{1-k_n^2})}{K(k_n)}= \sqrt n \]
    여기서 \(K(k)\)는 제1종타원적분 K (complete elliptic integral of the first kind)
  • complementary singular value  \(k'=\sqrt{1-k^2}\)
  • 자코비 세타함수를 이용하면, 복소상반평면에서 정의된 함수로 생각할 수 있는데 이 경우 다음이 만족된다\[k(\sqrt{-n})=k_n\]\[\frac{K'}{K}(k\left(\sqrt{-n})\right)= \sqrt n \]
  • 여러 문헌에서는 양수 \(r\)에 대하여 \(\lambda^{*}(r):=k(i\sqrt{r})\) 로 정의된 함수가 사용되기도 한다
  • 라마누잔이 하디에게 보낸 1913년의 편지에는 다음이 수록되어 있었다\[k_{210}=\left(-1+\sqrt{2}\right)^2 \left(2-\sqrt{3}\right) \left(8-3 \sqrt{7}\right) \left(-\sqrt{6}+\sqrt{7}\right)^2 \left(-3+\sqrt{10}\right)^2 \left(4-\sqrt{15}\right)^2 \left(-\sqrt{14}+\sqrt{15}\right) \left(6-\sqrt{35}\right)\]
  • 타원곡선complex multiplication 이론과 깊은 관계가 있다


자코비 세타함수와의 관계

  • 자코비 세타함수를 이용하여, 다음과 같이 표현 가능\[k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}\]\[k'=k'(\tau)=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}\]

 

 

special values

\(k(\sqrt{-1})=\frac{1}{\sqrt{2}}\)

\(k(\sqrt{-2})=-1+\sqrt{2}\)

\(k(\sqrt{-3})=\frac{\sqrt{6}-\sqrt{2}}{4}\)

\(k(\sqrt{-4})=3-2\sqrt{2}\)

 

제1종 타원적분과의 관계

\[K(\frac{1}{\sqrt{2}})=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\] \[K(\sqrt{2}-1)=\frac{\sqrt{\sqrt{2}+1}\Gamma(\frac{1}{8})\Gamma(\frac{3}{8})}{2^{13/4}\sqrt{\pi}}\] \[K\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{4\sqrt[4]{3}\sqrt{\pi}}=1.5981420\cdots\]\[K\left(3-2\sqrt{2}\right)=\frac{(2+\sqrt{2})\Gamma(\frac{1}{4})^2}{16\sqrt{\pi}}=1.58255\cdots\]

\[\frac{K'}{K}(\frac{1}{\sqrt{2}})= 1\] \[\frac{K'}{K}(\sqrt{2}-1)= \sqrt{2}\] \[\frac{K'}{K}\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)= \sqrt{3}\] \[\frac{K'}{K}\left(3-2\sqrt{2}\right)= \sqrt{4}\]

 

 

special values

  • 아래의 표는 \(r\)과 \(\lambda^{*}(r)=k(\sqrt{-r})\)의 값

\( \begin{array}{cc} 1 & \frac{1}{\sqrt{2}} \\ 2 & -1+\sqrt{2} \\ 3 & \frac{-1+\sqrt{3}}{2 \sqrt{2}} \\ 4 & 3-2 \sqrt{2} \\ 5 & \frac{1}{2} \left(-\sqrt{3-\sqrt{5}}+\sqrt{-1+\sqrt{5}}\right) \\ 6 & \left(2-\sqrt{3}\right) \left(-\sqrt{2}+\sqrt{3}\right) \\ 7 & \frac{3-\sqrt{7}}{4 \sqrt{2}} \\ 8 & \left(1+\sqrt{2}-\sqrt{2+2 \sqrt{2}}\right)^2 \\ 9 & \frac{1}{2} \left(\sqrt{2}-\sqrt[4]{3}\right) \left(-1+\sqrt{3}\right) \\ 10 & \left(-1+\sqrt{2}\right)^2 \left(-3+\sqrt{10}\right) \\ 11 & \frac{1}{2} \left(-\sqrt{\frac{11}{6}+\frac{2}{3 \sqrt[3]{17+3 \sqrt{33}}}-\frac{1}{3} \sqrt[3]{17+3 \sqrt{33}}}+\sqrt{\frac{1}{6}-\frac{2}{3 \sqrt[3]{17+3 \sqrt{33}}}+\frac{1}{3} \sqrt[3]{17+3 \sqrt{33}}}\right) \\ 12 & \left(-1+\sqrt{2}\right)^2 \left(-\sqrt{2}+\sqrt{3}\right)^2 \\ 13 & \frac{1}{2} \left(-\sqrt{19-5 \sqrt{13}}+\sqrt{-17+5 \sqrt{13}}\right) \\ 14 & -11-8 \sqrt{2}-4 \sqrt{5+4 \sqrt{2}}-2 \sqrt{2 \left(5+4 \sqrt{2}\right)}+2 \sqrt{11+8 \sqrt{2}}+2 \sqrt{2 \left(11+8 \sqrt{2}\right)}+\sqrt{2 \left(5+4 \sqrt{2}\right) \left(11+8 \sqrt{2}\right)} \\ 15 & \frac{\left(2-\sqrt{3}\right) \left(3-\sqrt{5}\right) \left(-\sqrt{3}+\sqrt{5}\right)}{8 \sqrt{2}} \\ 16 & \frac{\left(-1+\sqrt[4]{2}\right)^2}{\left(1+\sqrt[4]{2}\right)^2} \\ 17 & \frac{\sqrt{42+10 \sqrt{17}-13 \sqrt{\left(-3+\sqrt{17}\right) \left(5+\sqrt{17}\right)}-3 \sqrt{17 \left(-3+\sqrt{17}\right) \left(5+\sqrt{17}\right)}}-\sqrt{-38-10 \sqrt{17}+13 \sqrt{\left(-3+\sqrt{17}\right) \left(5+\sqrt{17}\right)}+3 \sqrt{17 \left(-3+\sqrt{17}\right) \left(5+\sqrt{17}\right)}}}{2 \sqrt{2}} \\ 18 & \left(-1+\sqrt{2}\right)^3 \left(2-\sqrt{3}\right)^2 \\ 21 & \frac{1}{2} \left(-\sqrt{1-\frac{1}{16} \left(3-\sqrt{7}\right)^2 \left(-\sqrt{3}+\sqrt{7}\right)^3}+\sqrt{1+\frac{1}{16} \left(3-\sqrt{7}\right)^2 \left(-\sqrt{3}+\sqrt{7}\right)^3}\right) \\ 22 & \left(10-3 \sqrt{11}\right) \left(-7 \sqrt{2}+3 \sqrt{11}\right) \\ 25 & \frac{\left(3-2 \sqrt[4]{5}\right) \left(-2+\sqrt{5}\right)}{\sqrt{2}} \\ 27 & \frac{1}{2} \left(-\sqrt{\frac{9}{2}+\sqrt[3]{2}-3 2^{2/3}}+\sqrt{-\frac{5}{2}-\sqrt[3]{2}+3 2^{2/3}}\right) \\ 30 & \left(2-\sqrt{3}\right) \left(-\sqrt{2}+\sqrt{3}\right)^2 \left(-\sqrt{5}+\sqrt{6}\right) \left(4-\sqrt{15}\right) \\ 33 & \frac{1}{2} \left(-\sqrt{-259+150 \sqrt{3}+78 \sqrt{11}-45 \sqrt{33}}+\sqrt{261-150 \sqrt{3}-78 \sqrt{11}+45 \sqrt{33}}\right) \\ 34 & \left(-1+\sqrt{2}\right)^2 \left(3 \sqrt{2}-\sqrt{17}\right) \left(-\sqrt{296+72 \sqrt{17}}+\sqrt{297+72 \sqrt{17}}\right) \\ 37 & \frac{1}{2} \left(-\sqrt{883-145 \sqrt{37}}+\sqrt{-881+145 \sqrt{37}}\right) \\ 42 & \left(-1+\sqrt{2}\right)^2 \left(2-\sqrt{3}\right)^2 \left(8-3 \sqrt{7}\right) \left(-\sqrt{6}+\sqrt{7}\right) \\ 45 & \frac{1}{2} \left(-\sqrt{1179+680 \sqrt{3}-527 \sqrt{5}-304 \sqrt{15}}+\sqrt{-1177-680 \sqrt{3}+527 \sqrt{5}+304 \sqrt{15}}\right) \\ 46 & \left(-18-13 \sqrt{2}+3 \sqrt{2 \left(147+104 \sqrt{2}\right)}-\sqrt{661+468 \sqrt{2}}\right) \left(18+13 \sqrt{2}+\sqrt{661+468 \sqrt{2}}\right) \\ 49 & \frac{1}{2} \left(-\sqrt{1-\frac{4096}{\left(\sqrt[4]{7}+\sqrt{4+\sqrt{7}}\right)^{12}}}+\sqrt{1+\frac{4096}{\left(\sqrt[4]{7}+\sqrt{4+\sqrt{7}}\right)^{12}}}\right) \\ 58 & \left(-1+\sqrt{2}\right)^6 \left(-99+13 \sqrt{58}\right) \\ 64 & \frac{\left(-2^{5/8}+\sqrt{1+\sqrt{2}}\right)^2}{\left(2^{5/8}+\sqrt{1+\sqrt{2}}\right)^2} \\ 210 & \left(-1+\sqrt{2}\right)^2 \left(2-\sqrt{3}\right) \left(8-3 \sqrt{7}\right) \left(-\sqrt{6}+\sqrt{7}\right)^2 \left(-3+\sqrt{10}\right)^2 \left(4-\sqrt{15}\right)^2 \left(-\sqrt{14}+\sqrt{15}\right) \left(6-\sqrt{35}\right) \\ 330 & \left(-1+\sqrt{2}\right)^2 \left(2-\sqrt{3}\right)^3 \left(-3+\sqrt{10}\right)^2 \left(10-3 \sqrt{11}\right) \left(3 \sqrt{5}-2 \sqrt{11}\right)^2 \left(4-\sqrt{15}\right) \left(-4 \sqrt{2}+\sqrt{33}\right)^2 \left(-3 \sqrt{6}+\sqrt{55}\right) \\ 462 & \left(2-\sqrt{3}\right)^2 \left(-\sqrt{2}+\sqrt{3}\right)^4 \left(8-3 \sqrt{7}\right)^2 \left(2 \sqrt{2}-\sqrt{7}\right)^2 \left(10-3 \sqrt{11}\right) \left(-7 \sqrt{2}+3 \sqrt{11}\right)^2 \left(-\sqrt{21}+\sqrt{22}\right) \left(76-5 \sqrt{231}\right) \end{array} \)

 

 


역사

 

 

메모

 

 

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수학용어번역

  • singular - 대한수학회 수학용어집

 


 

관련논문

  • Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominator." IMA J. Numerical Analysis 12, 519-526, 1992