Landen's transformation is a mapping of the parameters of an
elliptic integral
In integral calculus, 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 and Leonhard Euler (). Their name originates from their originally arising in ...
, useful for the efficient numerical evaluation of elliptic functions. It was originally due to
John Landen
John Landen (23 January 1719 – 15 January 1790) was an English mathematician.
Life
He was born at Peakirk, near Peterborough in Northamptonshire, on 28 January 1719. He was brought up to the business of a surveyor, and acted as land agent to ...
and independently rediscovered by
Carl Friedrich Gauss
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 ...
.
Statement
The
incomplete elliptic integral of the first kind is
:
where
is 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 ...
. Landen's transformation states that if
,
,
,
are such that
and
, then
:
Landen's transformation can similarly be expressed in terms of the elliptic modulus
and its complement
.
Complete elliptic integral
In Gauss's formulation, the value of the integral
:
is unchanged if
and
are replaced by their
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
and
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
s respectively, that is
:
:
Therefore,
:
:
From Landen's transformation we conclude
:
and
.
Proof
The transformation may be effected by
integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
. It is convenient to first cast the integral in an
algebraic form by a substitution of
,
giving
:
A further substitution of
gives the desired result
:
This latter step is facilitated by writing the radical as
:
and the infinitesimal as
:
so that the factor of
is recognized and cancelled between the two factors.
Arithmetic-geometric mean and Legendre's first integral
If the transformation is iterated a number of times, then the parameters
and
converge very rapidly to a common value, even if they are initially of different orders of magnitude. The limiting value is called the
arithmetic-geometric mean of
and
,
. In the limit, the integrand becomes a constant, so that integration is trivial
:
The integral may also be recognized as a multiple of
Legendre's complete elliptic integral of the first kind. Putting
:
Hence, for any
, the arithmetic-geometric mean and the complete elliptic integral of the first kind are related by
:
By performing an inverse transformation (reverse arithmetic-geometric mean iteration), that is
:
:
:
the relationship may be written as
:
which may be solved for the AGM of a pair of arbitrary arguments;
:
References
{{Reflist
* Louis V. King
On The Direct Numerical Calculation Of Elliptic Functions And Integrals' (Cambridge University Press, 1924)
Elliptic functions