Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form

 f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt

where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.

In general, elliptic integrals cannot be expressed in terms of elementary functions. Exceptions to this general rule are when P has repeated roots, or when R(x,y) contains no odd powers of y. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.

Contents

Argument notation

Elliptic integrals are a function of two arguments. These arguments are expressed in a variety of different but completely equivalent ways (they give the same elliptic integral). Most texts adhere to a canonical naming scheme, using the following naming conventions.

For expressing one argument:

  • α, the modular angle;
  • k = sin α, the elliptic modulus or eccentricity;
  • m = k2 = sin2α, the parameter.

Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably.

The other argument can likewise be expressed as φ, the amplitude, or as x or u, where x = sin φ = sn u and sn is one of the Jacobian elliptic functions.

Specifying the value of any one of these quantities determines the others. Note that u also depends on m. Some additional relationships involving u include

\cos \varphi = \textrm{cn} \; u, \qquad \textrm{and} \qquad \sqrt{1 - m \sin^2 \varphi} = \textrm{dn}\; u.

The latter is sometimes called the delta amplitude and written as Δ(φ) = dn u. Sometimes the literature also refers to the complementary parameter, the complementary modulus, or the complementary modular angle. These are further defined in the article on quarter periods.

Incomplete elliptic integral of the first kind

The incomplete elliptic integral of the first kind F is defined as

 F(\varphi,k) = F(\varphi \,|\, k^2) = F(\sin \varphi ; k) = \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}.

This is the trigonometric form of the integral; substituting t = sin θ, x = sin φ, one obtains Jacobi's form:

 F(x ; k) = \int_{0}^{x} \frac{dt}{\sqrt{(1 - t^2)(1 - k^2 t^2)}}.

Equivalently, in terms of the amplitude and modular angle one has:

 F(\varphi \setminus \alpha) = F(\varphi, \sin \alpha) = \int_0^\varphi \frac{d \theta}{\sqrt{1-(\sin \theta \sin \alpha)^2}}.

In our notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude:

 F(\varphi, \sin \alpha) = F(\varphi \,|\, \sin^2 \alpha) = F(\varphi \setminus \alpha) = F(\sin \varphi ; \sin \alpha).

This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of our notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

With x = sn(u;k) one has:

F(x;k) = u;

thus, the Jacobian elliptic functions are inverses to the elliptic integrals.

Notational variants

There are still other conventions for the notation of elliptic integrals employed in the literature. Thus, the notation with interchanged arguments, F(k,φ), is also often encountered; and similarly E(k,φ) for the integral of the second kind. Abramowitz and Stegun substitute the integral of the first kind, F(φ,k), for the argument φ in their definition of the integrals of the second and third kinds, unless this argument is followed by a backslash: i.e. E(F(φ,k) | k2) for our E(φ | k2). Moreover, their complete integrals employ the "parameter" k2 as argument in place of our modulus k, i.e. K(k2) rather than K(k). And the integral of the third kind defined by Gradshteyn and Ryzhik, Π(φ,n,k), puts the amplitude φ first and not the "characteristic" n.

Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind E in trigonometric form is

 E(\varphi,k) = E(\varphi \,|\,k^2) = E(\sin\varphi;k) = \int_0^\varphi \sqrt{1-k^2 \sin^2\theta}\, d\theta.

Substituting t = sin θ, x = sin φ, one obtains Jacobi's form:

 E(x;k) = \int_0^x \frac{\sqrt{1-k^2 t^2} }{\sqrt{1-t^2}}\, dt.

Equivalently, in terms of the amplitude and modular angle:

 E(\varphi \setminus \alpha) = E(\varphi, \sin \alpha) = \int_0^\varphi \sqrt{1-(\sin \theta \sin \alpha)^2} \,d\theta.

Relations with the Jacobi elliptic functions include

E(\mathrm{sn}(u ; k) ; k) = \int_0^u \mathrm{dn}^2 (w ; k) \, dw = u - k^2 \int_0^u \mathrm{sn}^2 (w ; k) \, dw = (1-k^2)u + k^2 \int_0^u \mathrm{cn}^2 (w ; k) \, dw.

Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind Π is

 \Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi  \frac{1}{1-n\sin^2 \theta}
\frac {d\theta}{\sqrt{1-(\sin\theta\sin \alpha)^2}}, or
 \Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2}
\frac{dt}{\sqrt{(1-m t^2)(1-t^2) }}.

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value \Pi(1; \tfrac \pi 2 \,|\,m)\,\! is infinite, for any m.

A relation with the Jacobian elliptic functions is

 \Pi(n; \,\mathrm{sn}(u;k); \,k) = \int_0^u \frac{dw} {1 - n \,\mathrm{sn}^2 (w;k)}.

Complete elliptic integral of the first kind

Elliptic Integrals are said to be 'complete' when the amplitude φ=π/2 and therefore x=1. The complete elliptic integral of the first kind K may thus be defined as

K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}},

or more compactly in terms of the incomplete integral of the first kind as

K(k) = F(\tfrac{\pi}{2},k) = F(1;k).

It can be expressed as a power series

K(k) = \frac{\pi}{2} \sum_{n=0}^\infty \left[\frac{(2n)!}{2^{2 n} (n!)^2}\right]^2 k^{2n} = \frac{\pi}{2} \sum_{n=0}^\infty [P_{2 n}(0)]^2 k^{2n},

where Pn is the Legendre polynomial, which is equivalent to

K(k) = \frac{\pi}{2}\left\{1 + \left(\frac{1}{2}\right)^2 k^{2} + \left(\frac{1 \cdot 3}{2 \cdot 4}\right)^2 k^{4} + \cdots + \left[\frac{\left(2n - 1\right)!!}{\left(2n\right)!!}\right]^2 k^{2n} + \cdots \right\},

where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

K(k) = \tfrac{\pi}{2} \,{}_2F_1 \left(\tfrac{1}{2}, \tfrac{1}{2}; 1; k^2\right).

The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed in terms of the arithmetic-geometric mean:

K(k) = \frac {\pi /2}{\mathrm{agm}(1-k,1+k)}.

Special values

K(0) = \tfrac {\pi} {2}
K(1) = \infty
K\left(\tfrac{\sqrt{2}}{2}\right) = \tfrac{1}{4 \sqrt{\pi}} \;\Gamma\left(\tfrac{1}{4}\right)^2
K\left(\tfrac{1}{4}(\sqrt{6} - \sqrt{2})\right) = 2^{-\frac 7 3} 3^{\frac 1 4} \pi^{-1} \Gamma\left(\tfrac 1 3\right)^3
K\left(\tfrac{1}{4}(\sqrt{6} + \sqrt{2})\right) = 2^{-\frac 7 3} 3^{\frac 3 4} \pi^{-1} \Gamma\left(\tfrac 1 3\right)^3

Relation to Jacobi θ-function

The ration to Jacobi's θ function is given by

K(k)= \frac \pi 2 \theta_3^2(q),

where the nome q is q(k)=\exp\left(-\pi \frac{K^\prime(k)}{K(k)}\right).

Asymptotic expressions

K(k) \approx \frac {\pi} {2} + \frac {\pi} {8} \frac {k^2} {1-k^2} - \frac {\pi} {16} \frac {k^4} {1-k^2}

This approximation has a relative precision better than 3×10-4 for k < 1/2. Keeping only the first two terms is correct to 0.01 precision for k < 1/2.

Derivative and differential equation

\frac{\mathrm{d}K(k)}{\mathrm{d}k} = \frac{E(k)}{k(1-k^2)}-\frac{K(k)}{k}
\frac {\mathrm{d}} {\mathrm{d}k} \left[ k (1-k^2) \frac {\mathrm{d}K(k)} {\mathrm{d}k} \right] = k K(k)

A second solution to this equation is K(\sqrt{1-k^2}).

Complete elliptic integral of the second kind

The complete elliptic integral of the second kind E describes the circumference of the ellipse. It may be defined as

E(k) = \int_0^{\pi/2}\sqrt {1-k^2 \sin^2\theta}\ d\theta = \int_0^1 \frac{\sqrt{1-k^2 t^2}}{\sqrt{1-t^2}} dt,

or more compactly in terms of the incomplete integral of the second kind as

E(k) = E(\tfrac{\pi}{2},k) = E(1;k).

It can be expressed as a power series

E(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left[\frac{(2n)!}{2^{2 n} (n!)^2}\right]^2 \frac{k^{2n}}{1-2 n},

which is equivalent to

E(k) = \frac{\pi}{2}\left\{1 - \left(\frac{1}{2}\right)^2 \frac{k^2}{1} - \left(\frac{1 \cdot 3}{2 \cdot 4}\right)^2 \frac{k^4}{3} - \cdots - \left[\frac{\left(2n - 1\right)!!}{\left(2n\right)!!}\right]^2 \frac{k^{2n}}{2 n-1} - \cdots \right\}.

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

E(k) = \tfrac{\pi}{2}  \,{}_2F_1 \left(\tfrac{1}{2}, -\tfrac{1}{2}; 1; k^2 \right).

Special values

E(0) = \tfrac \pi 2
E(1) = 1 \,\!
E\left( \tfrac{\sqrt{2}}{2} \right) = \pi^{\frac{3}{2}} \Gamma\left( \tfrac{1}{4} \right)^{-2} + \tfrac{1}{8 \sqrt{\pi}} \Gamma\left( \tfrac{1}{4} \right)^2
E\left(\tfrac{1}{4}(\sqrt{6} - \sqrt{2})\right) = 2^{\frac 1 3} 3^{-\frac 3 4} \pi^2 \Gamma\left(\tfrac 1 3\right)^{-3} + 2^{-\frac {10} 3} 3^{-\frac {1} 4} \pi^{-1} (\sqrt3 + 1) \Gamma\left(\tfrac 1 3\right)^3
E\left(\tfrac{1}{4}(\sqrt{6} + \sqrt{2})\right) = 2^{\frac 1 3} 3^{-\frac 1 4} \pi^2 \Gamma\left(\tfrac 1 3\right)^{-3} + 2^{-\frac {10} 3} 3^{\frac 1 4} \pi^{-1} (\sqrt3 - 1) \Gamma\left(\tfrac 1 3\right)^3

Derivative and differential equation

\frac{\mathrm{d}E(k)}{\mathrm{d}k}=\frac{E(k)-K(k)}{k}
(k^2-1) \frac {\mathrm{d}} {\mathrm{d}k} \left[ k \;\frac {\mathrm{d}E(k)} {\mathrm{d}k} \right] = k E(k)

A second solution to this equation is E(\sqrt{1-k^2}) - K(\sqrt{1-k^2}).

Complete elliptic integral of the third kind

The complete elliptic integral of the third kind Π can be defined as

\Pi(n,k) = \int_0^{\pi/2}\frac{d\theta}{(1-n\sin^2\theta)\sqrt {1-k^2 \sin^2\theta}}.

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic n,

\Pi'(n,k) = \int_0^{\pi/2}\frac{d\theta}{(1+n\sin^2\theta)\sqrt {1-k^2 \sin^2\theta}}.

Partial derivatives

\frac{\partial\Pi(n,k)}{\partial n}=
\frac{1}{2(k^2-n)(n-1)}\left(E(k)+\frac{1}{n}(k^2-n)K(k)+\frac{1}{n}(n^2-k^2)\Pi(n,k)\right)
\frac{\partial\Pi(n,k)}{\partial k}=
\frac{k}{n-k^2}\left(\frac{E(k)}{k^2-1}+\Pi(n,k)\right)

Functional relations

Legendre's relation:

 K(k) E\left(\sqrt{1-k^2}\right) + E(k) K\left(\sqrt{1-k^2}\right) - K(k) K\left(\sqrt{1-k^2}\right) = \frac \pi 2.

See also

References

External links


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