Landen Survey Dante v. Manna and Victor H. Moll

Landen transformations are maps on the coefficients of an integral that preserve its value. We present a brief survey of their appearance in the literature. To Henry, who provides inspiration, taste and friendship 1. In the beginning there was Gauss In the year 1985, one of us had the luxury of attending a graduate course on Elliptic Functions given by Henry McKean at the Courant Institute. Among the many beautiful results he described in his unique style, there was a calculation of Gauss: take two positive real numbers a and b, with a > b, and form a new pair by replacing a with the arithmetic mean (a+ b)/2 and b with the geometric mean √ ab. Then iterate: (1.1) an+1 = an + bn 2 , bn+1 = √ anbn starting with a0 = a and b0 = b. Gauss [29] was interested in the initial conditions a = 1 and b = √ 2. The iteration generates a sequence of algebraic numbers which rapidly become impossible to describe explicitly, for instance, (1.2) a3 = 1 23 ( (1 + 4 √ 2) + 2 √ 2 8 √ 2 √