The Carter and Falconer [4] map function has inverse M1() = 1 4 [tan1(2) + tanh1 (2)]. Prove that the map function satisfies the differential

The Carter and Falconer [4] map function has inverse M−1(θ) = 1 4

[tan−1(2θ) + tanh−1

(2θ)].

Prove that the map function satisfies the differential equation M

(d)=1 − 16M4(d)

with initial condition M(0) = 0.

Deduce from these facts that M(d)

arises from a stationary renewal model.