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In
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by
Giulio Fagnano Giulio Carlo, Count Fagnano, Marquis de Toschi (26 September 1682 — 18 May 1766) was an Italian mathematician. He was probably the first to direct attention to the theory of elliptic integrals. Fagnano was born in Senigallia (at the time spelled ...
and
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(). Their name originates from their originally arising in connection with the problem of finding the
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. Modern mathematics defines an "elliptic integral" as any
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
of its two arguments, is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. 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 In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms ...
. Additional insight into the theory of the elliptic integral may be gained through the study of the
Schwarz–Christoffel mapping In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard ...
. Historically,
elliptic functions In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
were discovered as inverse functions of elliptic integrals.


Argument notation

''Incomplete elliptic integrals'' are functions of two arguments; ''complete elliptic integrals'' are functions of a single argument. These arguments are expressed in a variety of different but 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: * , the ''
modular angle Angular eccentricity is one of many parameters which arise in the study of the ellipse or ellipsoid. It is denoted here by α (alpha). It may be defined in terms of the eccentricity, ''e'', or the aspect ratio, ''b/a'' (the ratio of the sem ...
'' * , the ''elliptic modulus'' or ''
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
'' * , 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 , the ''amplitude'', or as or , where and is one of the Jacobian elliptic functions. Specifying the value of any one of these quantities determines the others. Note that also depends on . Some additional relationships involving include \cos \varphi = \operatorname u, \quad \textrm \quad \sqrt = \operatorname u. The latter is sometimes called the ''delta amplitude'' and written as . 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 period In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are special functions that appear in the theory of elliptic functions. The quarter periods ''K'' and i''K'' ′ are given by :K(m)=\int_0^ \frac and :K'(m) ...
s. In this 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\left(\varphi \mid \sin^2 \alpha\right) = 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 the notation is compatible with that used in the reference book by
Abramowitz and Stegun ''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
and that used in the integral tables by
Gradshteyn and Ryzhik ''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''Table of Integrals, Series, and Products ...
. There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, , is often encountered; and similarly for the integral of the second kind.
Abramowitz and Stegun ''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
substitute the integral of the first kind, , for the argument in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. for . Moreover, their complete integrals employ the ''parameter'' as argument in place of the modulus , i.e. rather than . And the integral of the third kind defined by
Gradshteyn and Ryzhik ''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''Table of Integrals, Series, and Products ...
, , puts the amplitude first and not the "characteristic" . Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, Wolfram's
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
software and
Wolfram Alpha WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data. WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Mat ...
define the complete elliptic integral of the first kind in terms of the parameter , instead of the elliptic modulus .


Incomplete elliptic integral of the first kind

The incomplete elliptic integral of the first kind is defined as F(\varphi,k) = F\left(\varphi \mid k^2\right) = F(\sin \varphi ; k) = \int_0^\varphi \frac . This is the trigonometric form of the integral; substituting and , one obtains the Legendre normal form: F(x ; k) = \int_^ \frac. Equivalently, in terms of the amplitude and modular angle one has: F(\varphi \setminus \alpha) = F(\varphi, \sin \alpha) = \int_0^\varphi \frac. With one has: F(x;k) = u; demonstrating that this Jacobian elliptic function is a simple inverse of the incomplete elliptic integral of the first kind. The incomplete elliptic integral of the first kind has following addition theorem: F\bigl arctan(x),k\bigr+ F\bigl arctan(y),k\bigr= F\left arctan\left(\frac\right) + \arctan\left(\frac\right),k\right The elliptic modulus can be transformed that way: F\bigl arcsin(x),k\bigr= \fracF\left arcsin\left(\frac\right),\frac\right


Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind in trigonometric form is E(\varphi,k) = E\left(\varphi \,, \,k^2\right) = E(\sin\varphi;k) = \int_0^\varphi \sqrt\, d\theta. Substituting and , one obtains the Legendre normal form: E(x;k) = \int_0^x \frac\,dt. Equivalently, in terms of the amplitude and modular angle: E(\varphi \setminus \alpha) = E(\varphi, \sin \alpha) = \int_0^\varphi \sqrt \, d\theta. Relations with the
Jacobi elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tri ...
include \begin E\bigl(\operatorname(u ; k) ; k\bigr) &= \int_0^u \operatorname^2 (w ; k) \,dw \\ &= u - k^2 \int_0^u \operatorname^2 (w ; k) \,dw \\ &= \left(1-k^2\right)u + k^2 \int_0^u \operatorname^2 (w ; k) \,dw. \end The
meridian arc In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to de ...
length from the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
to
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
is written in terms of : m(\varphi) = a\left(E(\varphi,e)+\fracE(\varphi,e)\right), where is the
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
, and is the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
. The incomplete elliptic integral of the second kind has following addition theorem: E\bigl arctan(x),k\bigr+ E\bigl arctan(y),k\bigr= E\left arctan\left(\frac\right) + \arctan\left(\frac\right),k\right+ \frac\left(\frac+\frac\right) The elliptic modulus can be transformed that way: E\bigl arcsin(x),k\bigr= \left(1+\sqrt\right)E\left arcsin\left(\frac\right),\frac\right- \sqrtF\bigl arcsin(x),k\bigr+ \frac


Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind is \Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac \frac or \Pi(n ; \varphi \,, \,m) = \int_^ \frac \frac. The number is called the characteristic and can take on any value, independently of the other arguments. Note though that the value is infinite, for any . A relation with the Jacobian elliptic functions is \Pi\bigl(n; \,\operatorname(u;k); \,k\bigr) = \int_0^u \frac . The meridian arc length from the equator to latitude is also related to a special case of : m(\varphi)=a\left(1-e^2\right)\Pi\left(e^2 ; \varphi \,, \,e^2\right).


Complete elliptic integral of the first kind

Elliptic Integrals are said to be 'complete' when the amplitude and therefore . The complete elliptic integral of the first kind may thus be defined as K(k) = \int_0^\tfrac \frac = \int_0^1 \frac, or more compactly in terms of the incomplete integral of the first kind as K(k) = F\left(\tfrac,k\right) = F\left(\tfrac \,, \, k^2\right) = F(1;k). It can be expressed as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
K(k) = \frac\sum_^\infty \left(\frac\right)^2 k^ = \frac \sum_^\infty \bigl(P_(0)\bigr)^2 k^, where is the
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
, which is equivalent to K(k) = \frac\left(1+\left(\frac\right)^2 k^2+\left(\frac\right)^2 k^4+\cdots+\left(\frac\right)^2 k^+\cdots\right), where denotes the
double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...
. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as K(k) = \tfrac \,_2F_1 \left(\tfrac, \tfrac; 1; k^2\right). The complete elliptic integral of the first kind is sometimes called the
quarter period In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are special functions that appear in the theory of elliptic functions. The quarter periods ''K'' and i''K'' ′ are given by :K(m)=\int_0^ \frac and :K'(m) ...
. It can be computed very efficiently in terms of the arithmetic–geometric mean: K(k) = \frac. Therefore the modulus can be transformed that way: \begin K(k) &= \frac \\ pt& = \frac \\ pt&= \frac \\ pt& = \fracK\left(\frac\right) \end This expression is valid for all n \isin \mathbb and : K(k) = n\left sum_^ \operatorname\left(\fracK(k);k\right)\rightK\left ^n\prod_^\operatorname\left(\fracK(k);k\right)^2\right


Relation to the gamma function

If and r \isin \mathbb^+ (where is the modular lambda function), then is expressible in closed form in terms of the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. For example, and give, respectively, K\left(\sqrt-1\right)=\frac, and K\left(\frac\right)=\frac. More generally, the condition that \frac=\frac be in an
imaginary quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
can be analytically extended to the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
is sufficient. For instance, if , then and K\left(e^\right)=\frac.


Relation to Jacobi theta function

The relation to Jacobi's theta function is given by K(k)=\frac\theta_3^2(q), where the nome is q(k) = \exp\left(-\pi \frac\right).


Asymptotic expressions

K\left(k\right)\approx\frac+\frac\frac-\frac\frac This approximation has a relative precision better than for . Keeping only the first two terms is correct to 0.01 precision for .


Differential equation

The differential equation for the elliptic integral of the first kind is \frac\left(k\left(1-k^2\right)\frac\right) = k \, K(k) A second solution to this equation is . This solution satisfies the relation \fracK(k) = \frac-\frac.


Continued fraction

A
continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
expansion is: \frac = -\frac + \sum^_ \frac = -\frac + \cfrac, where the nome is .


Complete elliptic integral of the second kind

The complete elliptic integral of the second kind is defined as E(k) = \int_0^\tfrac \sqrt \, d\theta = \int_0^1 \frac \, dt, or more compactly in terms of the incomplete integral of the second kind as E(k) = E\left(\tfrac,k\right) = E(1;k). For an ellipse with semi-major axis and semi-minor axis and eccentricity , the complete elliptic integral of the second kind is equal to one quarter of the
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
of the ellipse measured in units of the semi-major axis . In other words: C = 4 a E(e). The complete elliptic integral of the second kind can be expressed as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
E(k) = \frac\sum_^\infty \left(\frac\right)^2 \frac, which is equivalent to E(k) = \frac\left(1-\left(\frac12\right)^2 \frac-\left(\frac\right)^2 \frac-\cdots-\left(\frac\right)^2 \frac-\cdots\right). In terms of the
Gauss hypergeometric function Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, the complete elliptic integral of the second kind can be expressed as E(k) = \tfrac \,_2F_1 \left(\tfrac12, -\tfrac12; 1; k^2 \right). The modulus can be transformed that way: E(k) = \left(1+\sqrt\right)\,E\left(\frac\right) - \sqrt\,K(k)


Computation

Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean. Define sequences and , where , and the recurrence relations , hold. Furthermore, define c_n=\sqrt. By definition, a_\infty = \lim_ a_n = \lim_ g_n = \operatorname\left(1, \sqrt\right). Also \lim_ c_n=0. Then E(k) = \frac\left(1-\sum_^ 2^ c_n^2\right). In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all . To speed up computation further, the relation can be used. Furthermore, if and r \isin \mathbb^+ (where is the modular lambda function), then is expressible in closed form in terms of K(k)=\frac and hence can be computed without the need for the infinite summation term. For example, and give, respectively, p. 26, 161 E\left(\frac\right)=K\left(\frac\right)+\frac, and E\left(\frac\right)=\fracK\left(\frac\right)+\frac.


Derivative and differential equation

\frac = \frac \left(k^2-1\right) \frac \left( k \;\frac \right) = k E(k) A second solution to this equation is .


Complete elliptic integral of the third kind

The complete elliptic integral of the third kind can be defined as \Pi(n,k) = \int_0^\frac \frac. Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the ''characteristic'' , \Pi'(n,k) = \int_0^\frac \frac. Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.


Partial derivatives

\begin \frac &= \frac\left(E(k)+\frac\left(k^2-n\right)K(k) + \frac \left(n^2-k^2\right)\Pi(n,k)\right) \\ pt \frac &= \frac\left(\frac+\Pi(n,k)\right) \end


Functional relations

Legendre's relation In mathematics, Legendre's relation can be expressed in either of two forms: as a relation between complete elliptic integrals, or as a relation between periods and quasiperiods of elliptic functions. The two forms are equivalent as the periods and ...
: K(k) E\left(\sqrt\right) + E(k) K\left(\sqrt\right) - K(k) K\left(\sqrt\right) = \frac \pi 2.


See also

*
Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
*
Schwarz–Christoffel mapping In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard ...
*
Carlson symmetric form In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms ...
*
Jacobi's elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tri ...
*
Weierstrass's elliptic functions In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by th ...
*
Jacobi theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
*
Ramanujan theta function In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant for ...
* Arithmetic–geometric mean * Pendulum period *
Meridian arc In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to de ...


References


Notes


References


Sources

* * * * * * * * * *


External links

*
Eric W. Weisstein, "Elliptic Integral" (Mathworld)Matlab code for elliptic integrals evaluation
by elliptic project

(Exstrom Laboratories)
A Brief History of Elliptic Integral Addition Theorems
{{Algebraic curves navbox Elliptic functions Special hypergeometric functions