Landen's Formula

$${\vartheta _3(z,t)\vartheta _4(z,t)\over \vartheta _4(2z,2t)... ...{\vartheta _2(z,t)\vartheta _1(z,t)\over \vartheta _1(2z,2t)},$$

where $\vartheta _i$ are Theta Functions. This transformation was used by Gauß to show that Elliptic Integrals could be computed using the Arithmetic-Geometric Mean.