There is a close and beautiful connection
between the transformation theory for elliptic integrals and the very rapid
approximation of 
. This connection was first made explicit by Ramanujan in
his 1914 paper ``Modular Equations and Approximations to
''[26]. We might emphasize that Algorithms 1 and 2 are not to
be found in Ramanujan's work, indeed no recursive approximation of 
is
considered, but as we shall see they are intimately related to his analysis.
Three central examples are:
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