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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">겔폰드-슈나이더 정리</h5> | ||
+ | 겔폰드-슈나이더 (1934) | ||
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+ | <math>\alpha \ne 0</math>,<math>\alpha \ne 1</math>,<math>\beta\notin \mathbb{Q}</math> 인 복소수 <math>\alpha</math>와 <math>\beta</math> 가 대수적수이면, <math>\alpha^{\beta} =\exp\{\beta \log \alpha\}</math> 는 초월수이다. | ||
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+ | '''Comments''' | ||
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+ | * In general, <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is [http://en.wikipedia.org/wiki/Multivalued_function multivalued], where "log" stands for the [http://en.wikipedia.org/wiki/Complex_logarithm complex logarithm]. This accounts for the phrase "any value of" in the theorem's statement. | ||
+ | * An equivalent formulation of the theorem is the following: if<math>\alpha</math> and <math>\gamma</math> are nonzero algebraic numbers, and we take any non-zero logarithm of <math>\alpha</math>, then<math>(\log \gamma)/(\log \alpha)</math> is either rational or transcendental. | ||
+ | * If the restriction that <math>\beta</math> be algebraic is removed, the statement does not remain true in general (choose <math>\alpha=3</math> and <math>\beta=\log 2/\log 3</math>, which is transcendental, then <math>\alpha^{\beta}=2</math> is algebraic). A characterization of the values for <math>\alpha</math> and <math>\beta</math> which yield a transcendental <math>\alpha^{\beta}</math> is not known. | ||
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+ | (wikipedia 의 [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem#Statement Gelfond–Schneider theorem 페이지]에서) | ||
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+ | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">겔폰드 상수</h5> | ||
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+ | * <math>e^\pi</math> 를 겔폰드 상수라 함<br> | ||
+ | * <math>e^\pi=(e^{i\pi})^{-i}=(-1)^{i}</math><br> | ||
+ | * 겔폰드 슈나이더 정리를 적용하면, 초월수임이 증명.<br> | ||
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+ | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">겔폰드-슈나이더 상수</h5> | ||
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+ | * <math>2^{\sqrt2}</math><br> | ||
+ | * 겔폰드 슈나이더 정리를 적용하면, 초월수임이 증명.<br> | ||
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+ | * [[수학사연표 (역사)|수학사연표]] | ||
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+ | |||
+ | * 도서내검색<br> | ||
+ | ** http://books.google.com/books?q= | ||
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+ | |||
+ | * [http://www.math.sc.edu/~filaseta/gradcourses/Math785/main785.html Transcendental number theory]<br> | ||
+ | ** Michael Filaseta | ||
+ | ** Lecture notes | ||
+ | ** [http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes7.pdf Lindemann's Theorem] | ||
+ | ** [http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes8.pdf The Gelfond-Schneider Theorem and Some Related Results] | ||
+ | * http://ko.wikipedia.org/wiki/ | ||
+ | * http://en.wikipedia.org/wiki/ | ||
+ | * http://www.wolframalpha.com/input/?i= | ||
+ | * http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q= | ||
+ | * http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7= | ||
+ | * 다음백과사전 http://enc.daum.net/dic100/search.do?q= | ||
+ | * [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집] | ||
+ | * [http://navercast.naver.com/science/list 네이버 오늘의과학] | ||
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+ | |||
+ | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
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+ | * [http://www.artchive.com/ http://www.artchive.com] | ||
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2009년 6월 26일 (금) 13:43 판
겔폰드-슈나이더 정리
겔폰드-슈나이더 (1934)
\(\alpha \ne 0\),\(\alpha \ne 1\),\(\beta\notin \mathbb{Q}\) 인 복소수 \(\alpha\)와 \(\beta\) 가 대수적수이면, \(\alpha^{\beta} =\exp\{\beta \log \alpha\}\) 는 초월수이다.
Comments
- In general, \(\alpha^{\beta} = \exp\{\beta \log \alpha\}\) is multivalued, where "log" stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement.
- An equivalent formulation of the theorem is the following: if\(\alpha\) and \(\gamma\) are nonzero algebraic numbers, and we take any non-zero logarithm of \(\alpha\), then\((\log \gamma)/(\log \alpha)\) is either rational or transcendental.
- If the restriction that \(\beta\) be algebraic is removed, the statement does not remain true in general (choose \(\alpha=3\) and \(\beta=\log 2/\log 3\), which is transcendental, then \(\alpha^{\beta}=2\) is algebraic). A characterization of the values for \(\alpha\) and \(\beta\) which yield a transcendental \(\alpha^{\beta}\) is not known.
(wikipedia 의 Gelfond–Schneider theorem 페이지에서)
겔폰드 상수
- \(e^\pi\) 를 겔폰드 상수라 함
- \(e^\pi=(e^{i\pi})^{-i}=(-1)^{i}\)
- 겔폰드 슈나이더 정리를 적용하면, 초월수임이 증명.
겔폰드-슈나이더 상수
- \(2^{\sqrt2}\)
- 겔폰드 슈나이더 정리를 적용하면, 초월수임이 증명.
상위 주제
하위페이지
재미있는 사실
많이 나오는 질문과 답변
- 네이버 지식인
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
관련된 고교수학 또는 대학수학
관련된 다른 주제들
관련도서 및 추천도서
- 도서내검색
- 도서검색
참고할만한 자료
- Transcendental number theory
- Michael Filaseta
- Lecture notes
- Lindemann's Theorem
- The Gelfond-Schneider Theorem and Some Related Results
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- 다음백과사전 http://enc.daum.net/dic100/search.do?q=
- 대한수학회 수학 학술 용어집
- 네이버 오늘의과학
관련기사
- 네이버 뉴스 검색 (키워드 수정)
- http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
- http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
- http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
- http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
- http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
블로그
- 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=겔폰드슈나이더
- 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
이미지 검색
- http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
- http://images.google.com/images?q=
- http://www.artchive.com