대수적 함수와 아벨적분

이 항목의 스프링노트 원문주소

• 대수적 함수와 아벨적분
  

 

 

개요

• 2차 이상의 다항식 \(f(x,y)=0\)에 의해 정의되는 x의 대수함수 y가 있을 때의 x와 y의 유리식 R(x, y)의 적분.
  

 

 

초등함수와 덧셈 정리

• 사인/아크사인함수 덧셈정리의 적분표현
  \(\sin \left(\theta_1+\theta_2\right)=\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2\)
  \(\arcsin x+\arcsin y=\arcsin (x\sqrt{1-y^2}+y\sqrt{1-x^2})\)
  \(\int_0^x{\frac{1}{\sqrt{1-x^2}}}dx+\int_0^y{\frac{1}{\sqrt{1-x^2}}}dx = \int_0^{x\sqrt{1-y^2}+y\sqrt{1-x^2}}{\frac{1}{\sqrt{1-x^2}}}dx \)
  
• 탄젠트/아크탄젠트 함수 덧셈정리의 적분표현
  \(\tan(\theta_1+\theta_2)=\frac{\tan\theta_1+\tan\theta_2}{1-\tan\theta_1\tan\theta_2}\)
  \(\arctan x+\arctan y = \arctan{\frac{x+y}{1-xy}}\)
  \(\int_0^x \frac{dx}{1+x^2} + \int_0^y \frac{dx}{1+x^2} = \int_0^{\frac{x+y}{1-xy}} \frac{dx}{1+x^2}\)
  
  

• 지수/로그함수 덧셈정리의 적분표현
  \(e^x e^y=e^{x+y}\)
  \(\ln x + \ln y=\ln xy\)
  \(\int_{1}^{x} \frac{dx}{x}+\int_{1}^{y} \frac{dx}{x} = \int_{1}^{xy} \frac{dx}{x}\)
  

 

 

타원적분과 덧셈정리

• 다음과 같은 형태의 적분을 타원적분이라 함
  

\(\int R(x,y)\,dx\)

여기서 \(R(x,y)\)는 \(x,y\)의 유리함수, \(y^2\)= 중근을 갖지 않는 \(x\)의 3차식 또는 4차식.

• 타원적분의 덧셈정리(오일러)
  \(p(x)=1+mx^2+nx^4\)일 때,
  \(\int_0^x{\frac{1}{\sqrt{p(x)}}}dx+\int_0^y{\frac{1}{\sqrt{p(x)}}}dx = \int_0^{B(x,y)}{\frac{1}{\sqrt{p(x)}}}dx\)
  여기서 \(B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}\)
  

• 타원적분 항목 참조
  

 

 

아벨-야코비 정리

• 아벨-야코비 정리

 

 

역사

• 수학사연표

 

 

메모

When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian functions.
How did this happen? In mathematics, as in other sciences, the same processes can be observed again and again. First, new questions arise, for internal or external reasons, and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and therefore require less foreknowledge - even if it is only a matter of axiomatics, or set theory, or some such thing.
Felix Klein (1849-1925), Development of Mathematics in the 19th Century, 1928

 

 

관련된 항목들

• 아벨-야코비 정리
  
• periods
  
• 타원곡선의 주기
  
• 아벨(1802 – 1829)
  
• 야코비(1804 – 1851)
  

 

 

수학용어번역

• http://www.google.com/dictionary?langpair=en%7Cko&q=
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Abel–Jacobi_map
• http://en.wikipedia.org/wiki/Abelian_integral
• http://mathworld.wolfram.com/AbelianIntegral.html

 

 

관련논문

• http://www.jstor.org/action/doBasicSearch?Query=
• http://dx.doi.org/

 

 

관련도서

• A. Markushevich, Introduction to the Classical Theory of Abelian Functions