Elliptic integral
Encyclopedia
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
which can be expressed in the form
where is a rational function
of its two arguments, is a polynomial
of degree 3 or 4 with no repeated roots, and is a constant.
In general, elliptic integrals cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of . 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.
For expressing one argument:, the modular angle; sin α}}, the elliptic modulus or eccentricity; 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 , 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 u include
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 periods.
This is the trigonometric form of the integral; substituting , , one obtains Jacobi
's form:
Equivalently, in terms of the amplitude and modular angle one has:
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:
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
one has:
thus, the Jacobian elliptic functions are inverses to the elliptic integrals.
.
Substituting , , one obtains Jacobi's form:
.
Equivalently, in terms of the amplitude and modular angle:
.
Relations with the Jacobi elliptic functions include
.
, or
.
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
.
,
or more compactly in terms of the incomplete integral of the first kind as
.
It can be expressed as a power series
,
where is the Legendre polynomial, which is equivalent to
,
where denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
.
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:
.
where the nome q is
.
This approximation has a relative precision better than for . Keeping only the first two terms is correct to 0.01 precision for .
A second solution to this equation is
.
,
or more compactly in terms of the incomplete integral of the second kind as
.
It can be expressed as a power series
,
which is equivalent to
.
In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
.
A second solution to this equation is
.
.
Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic ,
.
's relation:
.
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
of an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. They were first studied by Giulio Fagnano and Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
. Modern mathematics defines an "elliptic integral" as any function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
which can be expressed in the form
where is a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
of its two arguments, is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
of degree 3 or 4 with no repeated roots, and is a constant.
In general, elliptic integrals cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of . However, with the appropriate reduction formula
Integration by reduction formulae
Integration by reduction formulae can be used when we want to integrate a function raised to the power n. If we have such an integral we can establish a reduction formula which can be used to calculate the integral for any value of n....
, 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
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...
. 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.
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:, the modular angle; sin α}}, the elliptic modulus or eccentricity; 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 , 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 u include
-
.
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 periods.
Incomplete elliptic integral of the first kind
The incomplete elliptic integral of the first kind is defined as-
.
This is the trigonometric form of the integral; substituting , , one obtains Jacobi
Carl Gustav Jakob Jacobi
Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...
's form:
-
.
Equivalently, in terms of the amplitude and modular angle one has:
-
.
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:
-
.
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
Abramowitz and Stegun
Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards...
and that used in the integral tables by Gradshteyn and Ryzhik.
With
one has:-
;
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, , is also often encountered; and similarly for the integral of the second kind. Abramowitz and Stegun 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 backslash: i.e. for our . Moreover, their complete integrals employ the "parameter" as argument in place of our modulus , i.e. rather than . And the integral of the third kind defined by Gradshteyn and Ryzhik, , puts the amplitude first and not the "characteristic" .Incomplete elliptic integral of the second kind
The incomplete elliptic integral of the second kind in trigonometric form is.Substituting , , one obtains Jacobi's form:
.Equivalently, in terms of the amplitude and modular angle:
.Relations with the Jacobi elliptic functions include
.Incomplete elliptic integral of the third kind
The incomplete elliptic integral of the third kind is, or.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
.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,or more compactly in terms of the incomplete integral of the first kind as
.It can be expressed as a power series
,where is the Legendre polynomial, which is equivalent to
,where denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
.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:
.Special values
Relation to Jacobi θ-function
The ration to Jacobi's θ function is given bywhere the nome q is
.Asymptotic expressions
This approximation has a relative precision better than for . Keeping only the first two terms is correct to 0.01 precision for .
Derivative and differential equation
A second solution to this equation is
.Complete elliptic integral of the second kind
The complete elliptic integral of the second kind describes the circumference of the ellipse. It may be defined as,or more compactly in terms of the incomplete integral of the second kind as
.It can be expressed as a power series
,which is equivalent to
.In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
.Special values
Derivative and differential equation
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.Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic ,
.Partial derivatives
Functional relations
LegendreAdrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
's relation:
.See also
- Elliptic curveElliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
- Schwarz–Christoffel mapping
- Carlson symmetric formCarlson symmetric formIn 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...
- Jacobi's elliptic functionsJacobi's elliptic functionsIn mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications...
- Weierstrass's elliptic functionsWeierstrass's elliptic functionsIn mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass...
- Jacobi theta function
- Ramanujan theta functionRamanujan theta functionIn mathematics, particularly q-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 form when written in terms of the Ramanujan theta...