# 타원적분

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## 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,

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

## 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

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

## 덧셈공식

$\int_0^x{\frac{1}{\sqrt{1-x^4}}}dx+\int_0^y{\frac{1}{\sqrt{1-x^4}}}dx = \int_0^{A(x,y)}{\frac{1}{\sqrt{1-x^4}}}dx$ 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 $B(x,y)=\frac{x\sqrt{p(y)}+y\sqrt{p(x)}}{1-nx^2y^2}$

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'elliptic'}, {'LOWER': '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.