"대수적 함수와 아벨적분"의 두 판 사이의 차이

수학노트
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(사용자 2명의 중간 판 41개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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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.<br> 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.<br> Felix Klein (1849-1925), Development of Mathematics in the 19th Century, 1928
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==개요==
  
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*  2차 이상의 다항식 <math>f(x,y)=0</math>에 의해 <math>x</math>의 대수적함수 <math>y</math>가 정의
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*  이 때, 유리함수 <math>R(x, y)</math>에 대하여, <math>\int R(x,y)\,dx</math> 형태의 적분을 아벨적분이라 함
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*  컴팩트 리만 곡면위의 적분론
  
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<h5>간단한 소개</h5>
 
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=e%5Ex%20e%5Ey%3D%20e%5E%7Bx%2By%7D ]
 
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Csin%20%5Cleft%28x%2By%5Cright%29%3D%5Csin%20x%20%5Ccos%20y%20%2B%20%5Ccos%20x%20%5Csin%20y%5C ]
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==초등함수와 덧셈 정리==
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Ctan%28%5Ctheta_1%2B%5Ctheta_2%29%3D%5Cfrac%7B%5Ctan%5Ctheta_1%2B%5Ctan%5Ctheta_2%7D%7B1-%5Ctan%5Ctheta_1%5Ctan%5Ctheta_2%7D ]
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*  사인/아크사인함수 덧셈정리의 적분표현
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:<math>\sin \left(\theta_1+\theta_2\right)=\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2</math>:<math>\arcsin x+\arcsin y=\arcsin (x\sqrt{1-y^2}+y\sqrt{1-x^2})</math>:<math>\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 </math>
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*  탄젠트/아크탄젠트 함수 덧셈정리의 적분표현
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:<math>\tan(\theta_1+\theta_2)=\frac{\tan\theta_1+\tan\theta_2}{1-\tan\theta_1\tan\theta_2}</math>
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:<math>\arctan x+\arctan y = \arctan{\frac{x+y}{1-xy}}</math>
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:<math>\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}</math>
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*  지수/로그함수 덧셈정리의 적분표현
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:<math>e^x e^y=e^{x+y}</math>:<math>\ln x + \ln y=\ln xy</math>:<math>\int_{1}^{x} \frac{dx}{x}+\int_{1}^{y} \frac{dx}{x} = \int_{1}^{xy} \frac{dx}{x}</math>
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cln%20x%20%2B%20%5Cln%20y%3D%20%5Cln%20xy ]
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Carcsin%20x%2B%5Carcsin%20y%3D%5Carcsin%20%28x%5Csqrt%7B1-y%5E2%7D%2By%5Csqrt%7B1-x%5E2%7D%29 ]
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==타원적분과 덧셈정리==
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*  다음과 같은 형태의 적분을 타원적분이라 함
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:<math>\int R(x,y)\,dx</math>
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여기서 <math>R(x,y)</math>는 <math>x,y</math>의 유리함수, <math>y^2</math>= 중근을 갖지 않는 <math>x</math>의 3차식 또는 4차식.
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;타원적분의 덧셈정리(오일러)
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다항식 <math>p(x)=1+mx^2+nx^4</math>에 대하여 다음이 성립한다
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:<math>\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</math> 여기서 <math>B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}</math>
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* [[타원적분]] 항목 참조
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Carctan%20x%2B%5Carctan%20y%20%3D%20%5Carctan%7B%5Cfrac%7Bx%2By%7D%7B1-xy%7D%7D ]
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===아벨의 덧셈 정리===
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;정리 (아벨)
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임의의 자연수 <math>m</math>에 대하여, 다음 형태의 아벨 적분
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:<math>
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\int_0^{x_1}R(x,y)\,dx+\cdots+\int_0^{x_m}R(x,y)\,dx
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</math>
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을 <math>m</math>에 의존하지 않는 어떤 자연수 <math>p</math> 개의 적분, 즉
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:<math>
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\int_0^{z_1}R(x,y)\,dx+\cdots+\int_0^{z_p}R(x,y)\,dx
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</math>
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으로 쓸 수 있다. 여기서, <math>z_1, \cdots, z_p</math>는 <math>x_1,\cdots, x_m</math>의 대수적 함수와 적당한 초등함수의 합으로 표현된다
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* [[아벨-야코비 정리]]
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* [[종수(genus)와 오일러표수]]
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_0%5Ex%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E2%7D%7D%7Ddx%2B%5Cint_0%5Ey%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E2%7D%7D%7Ddx%20%3D%20%5Cint_0%5E%7Bx%5Csqrt%7B1-y%5E2%7D%2By%5Csqrt%7B1-x%5E2%7D%7D%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E2%7D%7D%7Ddx%20 ]
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==역사==
  
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* [[수학사 연표]]
  
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_%7B1%7D%5E%7Bx%7D%20%5Cfrac%7Bdx%7D%7Bx%7D%2B%5Cint_%7B1%7D%5E%7By%7D%20%5Cfrac%7Bdx%7D%7Bx%7D%20%3D%20%5Cint_%7B1%7D%5E%7Bxy%7D%20%5Cfrac%7Bdx%7D%7Bx%7D ]
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[http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_0%5Ex%20%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D%20%20%2B%20%5Cint_0%5Ey%20%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D%20%3D%20%20%5Cint_0%5E%7B%5Cfrac%7Bx%2By%7D%7B1-xy%7D%7D%20%5Cfrac%7Bdx%7D%7B1%2Bx%5E2%7D ]
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==메모==
  
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<h5>관련된 학부 과목과 미리 알고 있으면 좋은 것들</h5>
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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
  
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* http://www.abelprize.no/nedlastning/litteratur/houzel_the_work.pdf
  
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==관련된 항목들==
  
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<h5>관련된 대학원 과목</h5>
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* [[아벨-야코비 정리]]
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* [[periods]]
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* [[타원곡선의 주기]]
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* [[닐스 헨릭 아벨(1802 – 1829)|아벨(1802 – 1829)]]
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* [[야코비(1804 – 1851)]]
  
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==사전 형태의 자료==
  
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<h5>관련된 다른 주제들</h5>
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* http://ko.wikipedia.org/wiki/
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* [http://en.wikipedia.org/wiki/Abel%E2%80%93Jacobi_map http://en.wikipedia.org/wiki/Abel–Jacobi_map]
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* http://en.wikipedia.org/wiki/Abelian_integral
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* http://mathworld.wolfram.com/AbelianIntegral.html
  
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==관련도서==
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* Dedekind, Richard, and Heinrich Weber. 2012. Theory of Algebraic Functions of One Variable. Vol. 39. History of Mathematics. Providence, RI: American Mathematical Society.
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* A. Markushevich, [http://books.google.com/books?id=-kpCRuZPzTwC Introduction to the Classical Theory of Abelian Functions]
  
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<h5>표준적인 도서 및 추천도서</h5>
 
  
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
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==리뷰, 에세이, 강의노트==
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* Griffiths, Phillip. 2004. “The Legacy of Abel in Algebraic Geometry.” In The Legacy of Niels Henrik Abel, 179–205. Berlin: Springer. http://publications.ias.edu/sites/default/files/legacy.pdf
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* Kleiman, Steven L. 2004. “What Is Abel’s Theorem Anyway?” In The Legacy of Niels Henrik Abel, 395–440. Berlin: Springer. http://link.springer.com/chapter/10.1007%2F978-3-642-18908-1_11
  
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<h5>위키링크</h5>
 
  
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* http://en.wikipedia.org/wiki/
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==관련논문==
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* Griffiths, Phillip A. 1976. “Variations on a Theorem of Abel.” Inventiones Mathematicae 35: 321–390. http://publications.ias.edu/sites/default/files/variationsonatheorem.pdf
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* Barnum, Harry Huntington. 1910. “Abel’s Theorem and the Addition Formulae for Elliptic Integrals.” Annals of Mathematics 11 (3) (April 1): 103–114. doi:10.2307/1967324.
  
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<h5>참고할만한 자료</h5>
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[[분류:리만곡면론]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q4666729 Q4666729]
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===Spacy 패턴 목록===
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* [{'LOWER': 'abel'}, {'OP': '*'}, {'LOWER': 'jacobi'}, {'LEMMA': 'map'}]

2021년 2월 17일 (수) 05:01 기준 최신판

개요

  • 2차 이상의 다항식 \(f(x,y)=0\)에 의해 \(x\)의 대수적함수 \(y\)가 정의
  • 이 때, 유리함수 \(R(x, y)\)에 대하여, \(\int R(x,y)\,dx\) 형태의 적분을 아벨적분이라 함
  • 컴팩트 리만 곡면위의 적분론


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)

다항식 \(p(x)=1+mx^2+nx^4\)에 대하여 다음이 성립한다 \[\int_0^x{\frac{1}{\sqrt{p(x)}}}dx+\int_0^y{\frac{1}{\sqrt{p(x)}}}dx = \int_0^{B(x,y)}{\frac{1}{\sqrt{p(x)}}}dx\] 여기서 \(B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}\)


아벨의 덧셈 정리

정리 (아벨)

임의의 자연수 \(m\)에 대하여, 다음 형태의 아벨 적분 \[ \int_0^{x_1}R(x,y)\,dx+\cdots+\int_0^{x_m}R(x,y)\,dx \] 을 \(m\)에 의존하지 않는 어떤 자연수 \(p\) 개의 적분, 즉 \[ \int_0^{z_1}R(x,y)\,dx+\cdots+\int_0^{z_p}R(x,y)\,dx \] 으로 쓸 수 있다. 여기서, \(z_1, \cdots, z_p\)는 \(x_1,\cdots, x_m\)의 대수적 함수와 적당한 초등함수의 합으로 표현된다

역사



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


사전 형태의 자료


관련도서


리뷰, 에세이, 강의노트


관련논문

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'abel'}, {'OP': '*'}, {'LOWER': 'jacobi'}, {'LEMMA': 'map'}]
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