Thread: Integration of a step function

I've got a function

PV(t) = PV₀ * (1 + i_e)^{(D - t}) * (1 - i_i)^{(D - t)}

which feeds into a step function, and which I have to integrate

F(t) = ₀I^D PV₀ * (1 + i_e)^{(D - [t]}) * (1 - i_i)^{(D - [t])} dt

(₀I^D...Integral; [t]...ceiling t)

Hey mhcqr.

Have you considered breaking up the integral into analytic parts by partitioning it in the right way?

Hi chiro,

To be perfectly frank and honest, I haven't got the slightest clue any more; you see, it's more than two decades now since I've been heavily engaged with mathematics...

...how could such a breaking up potentially be looking like?

Cheers,
M.

As an example lets say we are integrating H(t-2) over the region 0 to 4 (H(t-a) is the Heaviside function where its 0 if t < a and 1 otherwise).

In this case you would break it up into two integrals: one with limits 0 to 2 and the other with limits 2 to 4 where the first integral would be 0dt and the other would be 1dt.

The basic idea is to make the integral analytic over the region so you can use the standard rules for that region.

So basically this would mean breaking up the integral into a sequence of consecutive steps, determine the result of each of these steps and finally summing these up?

Yep that's pretty much it.

Okay, so as long as I've got only a few interval steps to cover, it's kind of a piece of cake, but as the number of steps increases it appears to be rather a tedious undertaking, is it?

Considering the range t=0 to t=1,000,000 ideally I'd like to do F(1,000,000) - F(0) only...(I'm somehow more of the lazy persuasion!

You would probably want to use a computer if that is the case.

Whether the package is a symbolic one or numerical one is up to you.