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The sequence of odd numbers gives the continued fraction for Coth[1] ; the sequence of even numbers for BesselI[0, 1]/BesselI[1, 1] .

But for smaller e[s] one can show that
Abs[m[s]] (1 - Sinh[2 β ] -4 ) 1/8
where β can be deduced from
e[s] -(Coth[2 β ](1 + 2 EllipticK[4 Sech[2 β ] 2 Tanh[2 β ] 2 ] (-1 + 2 Tanh[2 β ] 2 )/ π ))
This implies that just below the critical point e 0 = - √ 2 (which corresponds to β = Log[1 + √ 2 ]/2 ) Abs[m] ~ (e 0 - e) 1/8 , where here 1/8 is a so-called critical exponent.