"타원적분의 singular value k"의 두 판 사이의 차이
| 9번째 줄: | 9번째 줄: | ||
* 자연수 <math>n </math> 에 대하여, 다음을 만족시키는 <math>k</math>를 singular value 라 한다<br><math>\frac{K'}{K}(k):=\frac{K(\sqrt{1-k^2})}{K(k)}= \sqrt n </math><br> | * 자연수 <math>n </math> 에 대하여, 다음을 만족시키는 <math>k</math>를 singular value 라 한다<br><math>\frac{K'}{K}(k):=\frac{K(\sqrt{1-k^2})}{K(k)}= \sqrt n </math><br> | ||
* complementary singular value <math>k'=\sqrt{1-k^2}</math> | * complementary singular value <math>k'=\sqrt{1-k^2}</math> | ||
| + | * [[자코비 세타함수]]를 이용하면, 복소상반평면에서 정의된 함수로 생각할 수 있다 | ||
| + | * 문헌에서는 양수 <math>r</math>에 대하여 <math>\lambda^{*}(r):=k(i\sqrt{r})</math> 로 정의된 함수로 표현되기도 한다 | ||
| 16번째 줄: | 18번째 줄: | ||
<h5 style="margin: 0px; line-height: 2em;">자코비 세타함수와의 관계</h5> | <h5 style="margin: 0px; line-height: 2em;">자코비 세타함수와의 관계</h5> | ||
| − | * [[자코비 세타함수]]<br><math>k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}</math><br><math>k'=k'(\tau)=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math><br> | + | * [[자코비 세타함수]]를 이용하여, 다음과 같이 표현 가능<br><math>k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}</math><br><math>k'=k'(\tau)=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math><br> |
| + | |||
| + | |||
| + | |||
| + | <h5>special values</h5> | ||
| + | |||
| + | * 아래의 표는 | ||
| + | |||
| + | <math>\lambda^{*}(r):=k(i\sqrt{r})</math> | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | <math> \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} </math> | ||
2009년 11월 29일 (일) 11:11 판
이 항목의 스프링노트 원문주소
개요
- 자연수 \(n \) 에 대하여, 다음을 만족시키는 \(k\)를 singular value 라 한다
\(\frac{K'}{K}(k):=\frac{K(\sqrt{1-k^2})}{K(k)}= \sqrt n \) - complementary singular value \(k'=\sqrt{1-k^2}\)
- 자코비 세타함수를 이용하면, 복소상반평면에서 정의된 함수로 생각할 수 있다
- 문헌에서는 양수 \(r\)에 대하여 \(\lambda^{*}(r):=k(i\sqrt{r})\) 로 정의된 함수로 표현되기도 한다
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
- 아래의 표는
\(\lambda^{*}(r):=k(i\sqrt{r})\)
\( \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} \)
재미있는 사실
역사
메모
관련된 항목들
수학용어번역
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문
관련도서 및 추천도서
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)