Landen's transformation explained

Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.^[1]

Statement

The incomplete elliptic integral of the first kind is

F(\varphi \setminus \alpha) = F(\varphi, \sin \alpha) = \int ^{\varphi} _{0} (d \theta)/(\sqrt{ 1- (\sin \theta \sin \alpha) ^{2}) },where \alpha is the modular angle. Landen's transformation states that if \alpha _{0} , \alpha _{1} , \varphi _{0} , \varphi _{1} are such that ( 1 + \sin\alpha _{1)(1} + \cos\alpha _{0)} = 2 and \tan(\varphi _{1} - \varphi _{0)} = \cos\alpha _{0} \tan \varphi _{0} , then\begin{align} F(\varphi _{0} \setminus \alpha _{0)} & = ( 1 + \cos\alpha ^{-1} _{0 )} F(\varphi _{1} \setminus \alpha _{1)} \\ & = \tfrac{ 1 }{ 2 }( 1 + \sin\alpha _{1)} F(\varphi _{1} \setminus \alpha _{1). \end{align}} Landen's transformation can similarly be expressed in terms of the elliptic modulus k = \sin\alpha and its complement k' = \cos\alpha .

Complete elliptic integral

In Gauss's formulation, the value of the integral

I = \int ^{(\pi)/(2)} _{0} (1)/(\sqrt{a ^{2} \cos ^{2(\theta)} + b ^{2} \sin ^{2(\theta)}) } \, d \thetais unchanged if a and b are replaced by their arithmetic and geometric means respectively, that isa _{1} = (a + b)/(2) , b _{1} = \sqrt{a b}, I _{1} = \int ^{(\pi)/(2)} ^{0} (1)/(\sqrt{a ^{2} _{1} \cos ^{2(\theta)} + b ^{2} _{1} \sin ^{2(\theta)}) } \, d \theta.Therefore,I = (1)/(K\left( a) (\sqrt{a ^{2} - b ^{2})/() }\right), I _{1= (2)/(a + b)} K\left( (a - b)/(a + b) \right). From Landen's transformation we concludeK\left( (\sqrt{a ^{2} - b ^{2})/() }\right)=\fracK\left(\frac\right) and I _{1=I} .

Proof

The transformation may be effected by integration by substitution. It is convenient to first cast the integral in an algebraic form by a substitution of

\theta = \arctan (x/b) , d \theta = (\cos ^{2} (\theta)/b) d x givingI = \int ^{(\pi)/(2)} ^{0} (1)/(\sqrt{a ^{2} \cos ^{2(\theta)} + b ^{2} \sin ^{2(\theta)}) } \, d \theta = \int _0^\infty \frac \, dx

A further substitution of

x = t + \sqrt{t ^{2} + a b} gives the desired result\begin{align}I & = \int ^{inf ty} ^{0} (1)/(\sqrt{(x ^{2} + a ^{2)} (x ^{2} + b ^{2)}) } \, dx \\ & = \int _^\infty \frac \, dt \\ & = \int _0^\infty\frac \, dt \end

This latter step is facilitated by writing the radical as

\sqrt{(x ^{2} + a ^{2)} (x ^{2} + b ^{2)}} = 2 x \sqrt{t ^{2} + \left( (a + b)/(2) \right) ^{2}}

and the infinitesimal as

dx = (x)/(\sqrt{t ^{2} + a b) } \, dt

so that the factor of

x 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

a and b 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 a and b , \operatorname{AGM}(a,b) . In the limit, the integrand becomes a constant, so that integration is trivialI = \int ^{(\pi)/(2)} ^{0} (1)/(\sqrt{a ^{2} \cos ^{2(\theta)} + b ^{2} \sin ^{2(\theta)}) } \, d\theta = \int _0^\frac \, d\theta = \frac

The integral may also be recognized as a multiple of Legendre's complete elliptic integral of the first kind. Putting

b ^{2} = a ^{2} ( 1 - k ^{2)} I = (1)/(a) \int ^{(\pi)/(2)} ^{0} (1)/(\sqrt{ 1- k ^{2} \sin ^{2(\theta)}) } \, d\theta = \frac F\left(\frac,k\right) = \frac K(k)

Hence, for any

a , the arithmetic-geometric mean and the complete elliptic integral of the first kind are related byK(k) = (\pi)/(2 \operatorname{AGM) ( 1 , \sqrt{ 1 - k ^{2})}}

By performing an inverse transformation (reverse arithmetic-geometric mean iteration), that is

a _{-1} = a + \sqrt{a ^{2} - b ^{2}} b _{-1} = a - \sqrt{a ^{2} - b ^{2}} \operatorname{AGM}(a,b) = \operatorname{AGM}\left(a + \sqrt{a ^{2} - b ^{2},a} - \sqrt{a ^{2} - b ^{2}\right)}

the relationship may be written as

K(k) = (\pi)/(2 \operatorname{AGM) ( 1 + k, 1 - k)}

which may be solved for the AGM of a pair of arbitrary arguments;

\operatorname{AGM}(u,v) = (\pi (u + v))/(4 K\left( (u - v)/(v + u) \right)) .

References

• Louis V. King On The Direct Numerical Calculation Of Elliptic Functions And Integrals (Cambridge University Press, 1924)

Notes and References

1. Gauss. C. F.. Nachlass. Arithmetisch geometrisches Mittel, Werke, Bd. 3. Königlichen Gesell. Wiss., Göttingen. 1876. 361–403.