Legendre form
Encyclopedia
In mathematics
, the Legendre forms of elliptic integral
s 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
(the ellipse being defined parametrically by 
, 
).
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric form
s. A more detailed treatment of the Legendre forms is given in the main article on elliptic integral
s.
the second kind as
and the third kind as
The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik as well as Numerical Recipes
. The choice of sign is that of Abramowitz and Stegun
as well as Gradshteyn and Ryzhik, but corresponds to the
of Numerical Recipes
.
The respective complete elliptic integrals are obtained by setting the amplitude,
, the upper limit of the integrals, to 
.
The Legendre form of an elliptic curve
is given by
's transformations. Descending Landen transformation decreases the modulus
towards zero, while increasing the amplitude 
. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of 
, zero or one, the integral is readily evaluated.
Most modern authors recommend evaluation in terms of the Carlson symmetric form
s, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library
and Numerical Recipes
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Legendre forms of elliptic integral
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...
s are a canonical set of three elliptic integrals to which all others may be reduced. Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
chose the name elliptic integrals because the second kind gives the arc length
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...
of unit semi-minor axis and eccentricity
Eccentricity (mathematics)
In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
(the ellipse being defined parametrically by , ).In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the 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...
s. A more detailed treatment of the Legendre forms is given in the main article on elliptic integral
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...
s.
Definition
The incomplete elliptic integral of the first kind is defined as,the second kind as
and the third kind as
The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik as well as Numerical Recipes
Numerical Recipes
Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul Teukolsky, William Vetterling and Brian Flannery. In various editions, the books have been in print since 1986...
. The choice of sign is that of 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...
as well as Gradshteyn and Ryzhik, but corresponds to the
of Numerical RecipesNumerical Recipes
Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul Teukolsky, William Vetterling and Brian Flannery. In various editions, the books have been in print since 1986...
.
The respective complete elliptic integrals are obtained by setting the amplitude,
, the upper limit of the integrals, to .The Legendre form of an elliptic curve
Elliptic curve
In 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...
is given by
Numerical Evaluation
The classic method of evaluation is by means of LandenJohn Landen
John Landen was an English mathematician,He was born at Peakirk near Peterborough in Northamptonshire, and died at Milton in the same county...
's transformations. Descending Landen transformation decreases the modulus
towards zero, while increasing the amplitude . Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of , zero or one, the integral is readily evaluated.Most modern authors recommend evaluation in terms of the 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...
s, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library
GNU Scientific Library
In computing, the GNU Scientific Library is a software library written in the C programming language for numerical calculations in applied mathematics and science...
and Numerical Recipes
Numerical Recipes
Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul Teukolsky, William Vetterling and Brian Flannery. In various editions, the books have been in print since 1986...
.