Complete Elliptic Integrals and the Arithmetic–Geometric Mean

This entry builds up two theories and connects them. The first consists of the complete elliptic integrals of the first and second kind \[ \begin{aligned} K(x) &= \int_0^{\frac{\pi}{2}} (1-x \sin(t)^2)^{-\frac{1}{2}}\,\text{d}t\\ E(x) &= \int_0^{\frac{\pi}{2}} (1-x \sin(t)^2)^{\frac{1}{2}}\,\text{d}t \end{aligned} \] for real or complex $x$.

The second one is the arithmetic-geometric mean function $\text{agm}(x,y)$, which is defined as the limit of the sequence obtained by replacing the pair $(x,y)$ with the pair consisting of the arithmetic and geometric means of $x$ and $y$, i.e. $(x,y)\mapsto (\frac{1}{2}(x+y), \sqrt{xy})$.

The two theories are then connected by proving (among other things) that: \[\text{agm}(a,b) = \frac{\pi a}{2K((a^2-b^2)/a^2)}\]

Various other important properties are shown, e.g.:

• Continuity, derivatives, antiderivatives of $K$ and $E$ as well as their relation to the hypergeometric function ${}_2 F_{1}$
• Legendre's identity \[K(x) E(1-x) + E(x) K(1-x) - K(x) K(1-x) = \frac{\pi}{2}\]
• The convergence of the AGM iterations, including uniform convergence and an estimate of the speed of convergence
• Upward and downward identities for $K$ and $E$, e.g. \[K(x^2) = \frac{K\left(\frac{4x}{(1+x)^2}\right)}{1+x}\]
• The relationship of the AGM to the Jacobi theta functions
• The Brent–Salamin algorithm to approximate $\pi$ via AGM iterations (abstractly, without rounding error analysis)