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In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
 of an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
. They were first studied by Giulio Fagnano and Leonhard Euler
Leonhard Euler

Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany.Euler made important discoveries in fields as diverse as calculus and graph theory....
. Modern mathematics defines an elliptic integral as any function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f which can be expressed in the form

where R 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....
 of its two arguments, P is the square root of a polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 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.






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In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
 of an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
. They were first studied by Giulio Fagnano and Leonhard Euler
Leonhard Euler

Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany.Euler made important discoveries in fields as diverse as calculus and graph theory....
. Modern mathematics defines an elliptic integral as any function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f which can be expressed in the form

where R 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....
 of its two arguments, P is the square root of a polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 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
Reduction formula

Reduction formula is a formula used to represent some expression in a simpler form.It may refer to:In mathematics:*decomposition of Multiple integral#Formulae of reduction...
, 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
Legendre form

In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi-minor axis and eccentricity ....
 and 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....
. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping
Schwarz-Christoffel mapping

In complex analysis, a discipline within mathematics, a Schwarz-Christoffel mapping is a transformation of the complex plane that maps the upper half-plane conformal maply to a polygon....
. Historically, elliptic functions were discovered as inverse functions of elliptic integrals. In particular, we have F (sn(z;k);k) = z, where sn is one of Jacobi's elliptic functions
Jacobi's elliptic functions

In 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 ....
.

Notation

Elliptic integrals are often expressed as functions for a variety of different arguments. These different arguments are completely equivalent (they give the same elliptic integral), but they 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:

  • the modular angle;
  • the elliptic modulus;
  • the parameter;


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

  • the amplitude
  • x where
  • u, where x = sn u and sn is one of the Jacobian elliptic functions


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

and

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
Quarter period

In mathematics, the quarter periods K and iK′ are special functions that appear in the theory of elliptic functions.The quarter periods K and iK' are given by...
s.

Incomplete elliptic integral of the first kind

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

Equivalently, using notation in Jacobi
Carl Gustav Jakob Jacobi

Carl Gustav Jacob Jacobi was a Prussian mathematician, widely considered to be the most inspiring teacher of his time and one of the greatest mathematicians of all time ....
's form, one sets ; then

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, , with the notations directly borrowed from the reference book of standards, Abramowitz and Stegun
Abramowitz and Stegun

Abramowitz and Stegun is the informal name of a mathematics reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards ....
. The use of the delimiters ; | \ is traditional in elliptic integrals.

However, there remain different conventions for the notation of elliptic integrals! The differences can be very confusing, especially to a novice . The functions that evaluate the elliptic integrals do not have standard and unique names and meanings (like sqrt
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
, sin
Sin

Sin is a term used mainly in a religion context to describe an act that violates a morality rule, or the state of having committed such a violation....
 and erf
Error function

In mathematics, the error function is a special function which occurs in probability, statistics, materials science, and partial differential equations....
 have). Even the literatures on the subject use differentiated notations. Gradstein, Ryzhik and the Wikipedia
Wikipedia

Wikipedia is a Free content, multilingualism encyclopedia project supported by the non-profit organization Wikimedia Foundation. Its name is a portmanteau of the words wiki and encyclopedia....
 article "Legendre form
Legendre form

In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi-minor axis and eccentricity ....
" use . The notation is equivalent to our ; also below.

Accordingly, if one translates the code from the Mathematica
Mathematica

Mathematica is a computational software program used widely in scientific, engineering, and mathematical fields and other areas of technical computing....
 language into the language used by Maple, one should replace the argument of the EllipticK function with its square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
. Correspondingly, in the translation from Maple to Mathematica, the argument should be replaced by its square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
. EllipticK(x) in Maple is almost equivalent to EllipticK[x^2] in Mathematica; one may expect to get the same result in both systems, at least while 0
Note that with u as defined above: thus, the Jacobian elliptic functions are inverses to the elliptic integrals.

Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind E is

Equivalently, using an alternate notation (substituting ),

Additional relations include

Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind is

or

or

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value is infinite, for any .

Complete elliptic integral of the first kind

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

or

It is a special case of the incomplete elliptic integral of the first kind:

The special case can be expressed as a power series
Power series

In mathematics, a power series is an infinite series of the formwhere an represents the coefficient of the nth term, c is a constant, and x varies around c ....


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
Quarter period

In mathematics, the quarter periods K and iK′ are special functions that appear in the theory of elliptic functions.The quarter periods K and iK' are given by...
. It can be computed in terms of the arithmetic-geometric mean
Arithmetic-geometric mean

In mathematics, the arithmetic-geometric mean of two positive real numbers x and y is defined as follows:First compute the arithmetic mean of x and y and call it a1....
.

Special values


The derivative of the complete elliptic integral of the first kind


Complete elliptic integral of the second kind


The complete elliptic integral of the second kind E may be defined as

or

It is a special case of the incomplete elliptic integral of the second kind:

that can be expressed as a power series
Power series

In mathematics, a power series is an infinite series of the formwhere an represents the coefficient of the nth term, c is a constant, and x varies around c ....


which is

In terms of the Gauss hypergeometric function
Hypergeometric series

In mathematics, a hypergeometric series, in the most general sense, is a power series in which the ratio of successive coefficients indexed by n is a rational function of n....
, the complete elliptic integral of the second kind can be expressed as

Special values




The derivative of the complete elliptic integral of the second kind


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 in , i.e.

The partial derivatives of the complete elliptic integral of the third kind



See also

  • Elliptic curve
    Elliptic curve

    In mathematics, an elliptic curve is a differentiable manifold, algebraic variety#Projective varieties algebraic curve of genus #Algebraic geometry one, on which there is a specified point O....
  • Schwarz-Christoffel map