겔폰드-슈나이더 정리

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.
  

 

 

 

 

 

하위페이지

• 0 토픽용템플릿
  
  • 0 상위주제템플릿
    

 

 

 

 

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관련된 다른 주제들

 

• 수학사연표

 

관련도서 및 추천도서

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• 도서검색
  
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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

 

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블로그

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• 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

 

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