Fourier series for Snowflake curve. E.g., www.tweedledum.com/rwg/cog.htm
Without the |abs|, the identity I gave yesterday is inf ==== n \ log(cot(2 x)) > -------------- = 2 %i x - 2 log(2 sin(x)), 0 < x < 2 pi, / n ==== 2 n = 0 or more generally, inf ==== n \ log(cot(2 x)) > -------------- = %i (%pi - 2 atan(cot(x))) / n ==== 2 n = 0 - 2 log(2 abs(sin(x))) . I'm surprised at the simplicity of the imagpart, whose source is the "random" subset of the cots which go negative as 2^n x galumphs off to +-infinity. rwg>Further massaging finally gives the very quadratically
convergent product, E.g.,
1 inf -- /===\ n | | 2 n - 2 x 2 (d5) | | tanh (2 x) = (1 - %e ) | | n = 0 (c6) apply_nouns(subst(3,inf,%)) 1/4 1/8 (d6) tanh(x) sqrt(tanh(2 x)) tanh (4 x) tanh (8 x) = - 2 x 2 (1 - %e ) (c10) taylor(trigexpand(subst(log(y),x,d6)),y,inf,33) 2 1 1 2 1 (d10)/T/ 1 - -- + -- + ----- + . . . = 1 - -- + -- + . . . 2 4 32 2 4 y y 8 y y y --rwg PYROCHEMICAL MICROCEPHALY