타원적분

개요

• 먼저 타원적분론 입문 참조
• \(R(x,y)\)는 \(x,y\)의 유리함수이고, \(y^2\)은 \(x\)의 3차 또는 4차식\[\int R(x,\sqrt{ax^3+bx^2+cx+d}) \,dx\] 또는\[\int R(x,\sqrt{ax^4+bx^3+cx^2+dx+e}) \,dx\]

타원 둘레의 길이

• 역사적으로 타원 둘레의 길이를 구하는 적분에서 그 이름이 기원함.
• 타원 \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)의 둘레의 길이는 \(4aE(k)\) 로 주어짐.\[k=\sqrt{1-\frac{b^2}{a^2}}\]\[E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^2\sin^2 \theta} d\theta =\int_{0}^{1}\frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}} dx=\int_{0}^{1}\frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx\]

정의

• 일반적으로 다음과 같은 형태로 주어지는 적분을 타원적분이라 부름

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

• 예를 들자면,

\[\int \frac{dx}{\sqrt{1-x^4}}\] \[\int \frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx\]

일종타원적분과 이종타원적분

• 제1종타원적분 K (complete elliptic integral of the first kind)\[K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}=\int_0^1\frac{1}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx\]\[K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\]
• 제2종타원적분 E (complete elliptic integral of the second kind)\[E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^2\sin^2 \theta} d\theta =\int_{0}^{1}\frac{\sqrt{1-k^2x^2}}{\sqrt{1-x^2}} dx=\int_{0}^{1}\frac{1-k^2x^2}{\sqrt{(1-x^2)(1-k^2x^2)}}\,dx\]\[E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)\]
• \(\,_2F_1(a,b;c;z)\)는 초기하급수(Hypergeometric series)

르장드르의 항등식

• 일종타원적분과 이종타원적분 사이에는 다음과 같은 관계가 성립

\[E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}\]

또는 \(\theta+\phi=\frac{\pi}{2}\) 에 대하여 \[E(\sin\theta)K(\sin\phi)+E(\sin\phi)K(\sin\theta)-K(\sin\theta)K(\sin\phi)=\frac{\pi}{2}\]

• 특별히 다음과 같은 관계가 성립함

\[2K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})-K(\frac{1}{\sqrt{2}})^2=\frac{\pi}{2}\]

산술기하평균함수(AGM)와 파이값의 계산에 응용

덧셈공식

• 파그나노의 공식 (렘니스케이트 곡선과 Lemniscatomy 항목 참조)

\[\int_0^x{\frac{1}{\sqrt{1-x^4}}}dx+\int_0^y{\frac{1}{\sqrt{1-x^4}}}dx = \int_0^{A(x,y)}{\frac{1}{\sqrt{1-x^4}}}dx\] 여기서 \(A(x,y)=\frac{x\sqrt{1-y^4}+y\sqrt{1-x^4}}{1+x^2y^2}\)

• 오일러의 일반화

\(p(x)=1+mx^2+nx^4\)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 \[B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}\]

메모

• 타원적분의 응용으로 단진자의 주기와 타원적분 항목 참조

관련된 항목들

• 타원곡선
• 타원함수
  • 바이어슈트라스 타원함수 ℘
• 자코비 세타함수
• 초기하급수(Hypergeometric series)
• 대수적 함수와 아벨적분
• 오일러치환

사전 형태의 자료

• http://ko.wikipedia.org/wiki/타원적분
• http://en.wikipedia.org/wiki/Elliptic_integral
• NIST Digital Library of Mathematical Functions
  • Chapter 19 Elliptic Integrals

관련논문

• In Search of the "Birthday" of Elliptic Functions - Bit by bit, the discoverers decided what it was they had discovered.
  • Rice, Adrian, 48-57, The Mathematical Intelligencer, Volume 30, Number 2, 2008-3
• Totaro, Burt. 2007. “Euler and Algebraic Geometry.” Bulletin of the American Mathematical Society 44 (4): 541–559. doi:10.1090/S0273-0979-07-01178-0.
• The Lemniscate and Fagnano's Contributions to Elliptic Integrals
  • AYOUB R
• A property of Euler's elastic curve
• The story of Landen, the hyperbola and the ellipse
  • Elemente der Mathematik, Volume 57, Number 1 / 2002년 2월
• Three Fermat Trails to Elliptic Curves
  • Ezra Brown, The College Mathematics Journal, Vol. 31, No. 3 (May, 2000), pp. 162-172
• Elliptic Curves
  • John Stillwell, The American Mathematical Monthly, Vol. 102, No. 9 (Nov., 1995), pp. 831-837
• Abel's Theorem on the Lemniscate
  • Michael Rosen, The American Mathematical Monthly, Vol. 88, No. 6 (Jun. - Jul., 1981), pp. 387-395

관련도서

• Elliptic functions and elliptic integrals
  • Viktor Prasolov, Yuri Solovyev
• Pi and the AGM
  • Jonathan M. Borwein, Peter B. Borwein

블로그

• 삼각치환에서 타원적분으로 피타고라스의 창, 2009-8-19

메타데이터

위키데이터

• ID : Q1126603

Spacy 패턴 목록

• [{'LOWER': 'elliptic'}, {'LEMMA': 'integral'}]

노트

말뭉치

1. For instance, while the arc length of a circle is given as a simple function of the parameter, computing the arc length of an ellipse requires an elliptic integral.
2. An elliptic integral is written when the parameter is used, when the elliptic modulus is used, and when the modular angle is used.
3. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping.
4. These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral).
5. This is referred to as the incomplete Legendre elliptic integral.
6. The integral is also called Legendres form for the elliptic integral of the first kind.
7. = 0 (1+2)122 , 0 < < 1, 0, also called Legendres form for the elliptic integral of the third kind.
8. The upper limit x in the Jacobi form of the elliptic integral of the first kind is related to the upper limit in the Legendre form by = sin .
9. returns values of the complete elliptic integral E(K).
10. returns values of the complete elliptic integral E(M).
11. returns values of the complete elliptic integral F(K).
12. returns values of the complete elliptic integral F(M).
13. Any elliptic integral can be expressed as a sum of elementary functions and linear combinations of canonical elliptic integrals of the first, second and third kinds.
14. TOMS577, a C++ library which evaluates Carlson's elliptic integral functions RC, RD, RF and RJ.
15. The point here is simply show one of the uses of the elliptic integral of the first kind.
16. However, the complete elliptic integral is taken from 0 to 90 degrees, but since that is exactly 1/4 over the total arc length due to symmetry.
17. Complete elliptic integral of the second kind Math.
18. Complete elliptic integral of the first kind.
19. Complete elliptic integral of the third kind.
20. Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively.
21. dy 1 + a cos x + b cos y (1) Keywords: Double elliptic integral, hypergeometric function 1 .
22. v i X r a SHARP DOUBLE INEQUALITY FOR COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND QI BAO r2 sin2 t)1/2dt is known as the Abstract.
23. Abstract We prove a simple relation for a special case of Carlsons elliptic integral RD.
24. Let K be the complete elliptic integral of the rst kind.