In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler.

In the modern definition, an elliptic integral is any function f which can be expressed in the form

$f(x)\; =\; \backslash int\_\{c\}^\{x\}\; R(t,P(t))\backslash \; dt$

where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 (a cubic or quartic) with no repeated roots, and c is a constant.

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

Besides the forms given below, the elliptic integrals may also be expressed in Legendre form and Carlson symmetric form. Additional insight into the theory of the indefinite integral may be gained through the study of the Schwarz-Christoffel mapping.

Notation

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Elliptic integrals are often expressed as functions of a variety of different arguments. These different arguments are completely equivalent (they give the same elliptic integral), but can be confusing due to their different appearance. Most texts adhere to a canonical naming scheme. Before defining the integrals, we review the naming conventions for the arguments:

• k the elliptic modulus
• m=k² the parameter
• $\backslash alpha$ the modular angle, $k=\backslash sin\; \backslash alpha$

Note that the above three are completely determined by one another; specifying one is the same as specifying another. The elliptic integrals will also depend on another argument; this can also be specified in a number of different ways:

• $\backslash phi$ the amplitude
• x where $x=\backslash sin\; \backslash phi=\; \backslash textrm\{sn\}\; \backslash ;\; u$
• u, where x=sn u and sn is one of the Jacobian elliptic functions

Specifying any one of these determines the others, and thus again, these may be used interchangeably in the notation. Note that u also depends on m. Some additional relationships involving u include

$\backslash cos\; \backslash phi\; =\; \backslash textrm\{cn\}\backslash ;\; u$

and

$\backslash sqrt\{1-m\backslash sin^2\; \backslash phi\}\; =\; \backslash textrm\{dn\}\backslash ;\; u$.

The latter is sometimes called the delta amplitude and written as $\backslash Delta(\backslash phi)=\backslash textrm\{dn\}\backslash ;\; u$.

Sometimes the literature 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

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The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as

$F(x;k)\; =$

\int_{0}^{x} \frac{1}{ \sqrt{(1-t^2)(1-k^2 t^2)} }\ dt \,\! Equivalently, using alternate notation above, one sets $x=\backslash sin\; \backslash phi\; ~,~\; t=\backslash sin\; \backslash theta\; ~,~\; =\backslash sin(\backslash alpha)\; ~,~\; m=k^2~\; \backslash ;\backslash !$; then

$F(x;k)\; =\; F(\backslash phi|m)\; =\; F(\backslash phi\backslash setminus\; \backslash alpha\; )\; =$

\int_0^\phi \frac{1}{ \sqrt{1-\sin^2 \alpha \sin^2 \theta}} \ d\theta \,\! where it is understood that when there is a vertical bar used, the argument following the vertical bar is the parameter (as defined above), and, when a backslash is used, it is followed by the modular angle. In this sense, $F(\backslash sin\; \backslash phi;\backslash sin\; \backslash alpha)\; =\; F(\backslash phi|\backslash sin\; (\backslash alpha\; )^2)\; =\; F(\backslash phi\backslash setminus\; \backslash alpha\; )~\; \backslash ,\backslash !$. These notations are borrowed from the book Abramowitz and Stegun; the use of the delimiters ; | \ is traditional in elliptic integrals.

There are differing conventions regarding notation of elliptic integrals. The differences can be very confusing, especially to a novice, see //en.wikipedia.org/wiki/Talk%3AElliptic_integral. The functions, which evaluate the elliptic integrals, do not have standard and unic names and meanings (like sqrt, sin and erf have). Diffent notations are used in the literature. Gradstein, Ryzhik ^*, Eq.(8.111)] and the wiki article Legendre form use notation $F(\backslash phi,k)\; \backslash ,\backslash !$, which is equivalent to our $F(\backslash phi|k^2)~\; \backslash ,\backslash !$; also $E(\backslash phi,k)=E(\backslash phi|k^2)~\; \backslash ,\backslash !$ below. For example, if one translates a code from langguage of Mathematica into Maple computer algebra system, one should replace the argument of the EllipticK function to its square (and to its sqrt at the translation from Maple to Mathematica), EllipticK(x) in Maple is almost equivalent of EllipticK^* in Mathematica; one may expect to get the same result in both systems, at least while 0

Note that

$F(x;k)\; =\; u\; \backslash ,\backslash !$

with u as defined above: thus, the Jacobian elliptic functions are inverses to the elliptic integrals.

Incomplete elliptic integral of the second kind

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The incomplete elliptic integral of the second kind E is

$E(x;k)\; =\; \backslash int\_\{0\}^\{x\}\; \backslash frac\{\; \backslash sqrt\{1-k^2\; t^2\}\; \}\{\; \backslash sqrt\{1-t^2\}\; \}\backslash \; dt$

Equivalently, using alternate notation, (substituting $t=\backslash sin\backslash theta$)

$E(x;k)\; =\; E(\backslash phi|m)\; =\; E(\backslash phi\backslash setminus\; \backslash alpha\; )\; =$

\int_0^\phi \sqrt{1-\sin^2 \alpha \sin^2 \theta} \ d\theta

Additional relations include

$E(\backslash phi|m)\; =\; \backslash int\_0^u\; \backslash textrm\{dn\}^2\; w\; \backslash ;dw\; =$

u-m\int_0^u \textrm{sn}^2 w \;dw = (1-m)u+m\int_0^u \textrm{cn}^2 w \;dw

Incomplete elliptic integral of the third kind

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The incomplete elliptic integral of the third kind $\backslash Pi$ is

$\backslash Pi(n;\; \backslash phi|m)\; =\; \backslash int\_\{0\}^\{x\}\; \backslash frac\{1\}\{1-nt^2\}$

\frac{1} {\sqrt{(1-k^2 t^2)(1-t^2) }}\ dt or

$\backslash Pi(n;\; \backslash phi|m)\; =\; \backslash int\_0^\backslash phi\; \backslash frac\{1\}\{1-n\backslash sin^2\; \backslash theta\}$

\frac {1}{\sqrt{ (1-\sin^2 \alpha \sin^2 \theta) }} \ d\theta or

$\backslash Pi(n;\; \backslash phi|m)\; =\; \backslash int\_0^u\; \backslash frac\{1\}\{1-n\; \backslash textrm\{sn\}^2\; (w|m)\}\; \backslash ;\; dw$

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

Complete elliptic integral of the first kind

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The complete elliptic integral of the first kind K is defined as

$K(k)\; =\; \backslash int\_\{0\}^\{1\}\; \backslash frac\{1\}\{\; \backslash sqrt\{(1-t^2)(1-k^2\; t^2)\}\; \}\backslash \; dt$

and can be computed in terms of the arithmetic-geometric mean.

It can also be calculated as

$K(k)\; =\; \backslash frac\{\backslash pi\}\{2\}\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; k^\{2n\}\; \backslash frac\{(2n)!(2n)!\}\{16^n\; n!n!n!n!\}$

Or in the form of an integral of the sine, when 0 ≤ k ≤ 1

$K(\; k\; )\; =\; \backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash frac\{d\backslash theta\}\{\backslash sqrt\; \{1\; -\; k^2\; \backslash sin\; ^2\; \backslash theta\; \}\}$

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

$K(k)\; =\; \backslash frac\{\backslash pi\}\{2\}\backslash left\backslash \{1\; +\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^2\; k^\{2\}\; +\; \backslash left(\; \backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\backslash right)^2\; k^\{4\}\; +\; \backslash ldots\; +\; \backslash left-\; 1\backslash right)!!\}\{2^\{n\}n!\}\backslash right^2\; k^\{2n\}\; +\; \backslash ldots\; \backslash right\backslash \}\; =\; \backslash frac\{\backslash pi\}\{2\}\; F\; \backslash left(\backslash frac\{1\}\{2\},\; \backslash frac\{1\}\{2\};\; 1\; ;\; k^\{2\}\backslash right)$

The complete elliptic integral of the first kind is sometimes called the quarter period.

Complete elliptic integral of the second kind

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The complete elliptic integral of the second kind E is defined as

$E(k)\; =\; \backslash int\_\{0\}^\{1\}\; \backslash frac\{\; \backslash sqrt\{1-k^2\; t^2\}\; \}\{\; \backslash sqrt\{1-t^2\}\; \}\backslash \; dt$

Or if 0 ≤ k ≤ 1:

$E(\; k\; )\; =\; \backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash sqrt\; \{1\; -\; k^2\; \backslash sin\; ^2\; \backslash theta\}\backslash \; d\backslash theta$

History

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Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular: we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.

See also

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• Elliptic curve
• Legendre form
• Carlson symmetric form
• Schwarz-Christoffel map

References

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• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See chapter 17).

Elliptic functions | Special hypergeometric functions

Elliptisches Integral | Integral elíptica | Intégrale elliptique | Integrale ellittico | Całki eliptyczne