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Good question - I assume so. I think they are actuarial formulae and having given it a google, a appears to be the accumulated amount function where a(t)=(1+i)^t (which is basically compound interest I think?), except that doesn't lead to a"(t)= ln(1+i)... unless I'm being silly. I have asked.
@hamrack Ah! I had tried a substitution but have never come across the Product Log function before. That is super interesting and v useful. Pretty sure numerical only would not be an issue here. Thanks!
crow-
It's kind of close to: t = ln (A(t)/A(0))/ln(1 + i) which is the formula for time for some compounding investment to hit a value of A(t), starting value A(0), with interest i. But the numerator of that fraction doesn't make a lot of sense here.
In other words the question might be "how long is it before my investment is worth e^(1 - n/(1 + i)) times what I started with?" Unusual.
Oh and a''(t) = ln(1+i) doesn't seem to be something that would occur in the real world either, I agree
frankenbike
Is รค the second derivative, and if so, the second derivative with respect to what?