Hi Derek,
The exact expression is indeed the acosh expression used in Matlab, and we would expect no less. The log expression is an approximation, and it's a bit of a holdover from the times when scientific software was not readily available. acosh was not so easy to calculate, but log tables were common. Leaving out the factor of pi*epsilon, the approximation goes as follows. Using the variable Cinv for 1/C, then
Cinv = acosh(D/2a)
so
cosh(Cinv) = D/2a
(exp(Cinv) + exp(-Cinv))/2 = D/2a
Now if D/2a is large, then Cinv is large and you can drop the exp(-Cinv) term compared to the exp(Cinv) term. Then
exp(Cinv) = D/a
Cinv = log(D/a)
The natural question is, how large is large?
Da = 2:.001:20; % D over a
Cinv1 = acosh(Da/2);
Cinv2 = log(Da);
plot(Da,Cinv1,Da,Cinv2)
legend('acosh(d/2a)','log(D/a)','location','northwest')
The plot shows a pretty good approximation for D/a greater than about 5.
