Notes: A family of ramp functions and the Beta function

Sometimes in a numerical method, you need to be able to continuously turn a calculation on or off in space or time (here I will pretend it’s in time). This can be easily accomplished if you have a function that starts at a value of 0 before some first time $t_0$, and rises up to a value of 1 by time $t_1$. Through an affine transformation you can always map $[t_0, t_1] \to [0, 1]$. An example “ramp” function is plotted above.

Now if this function appears in a differential equation, and you are integrating it with an $n^{\textrm{th}}$ order method, then it’s not enough for the function to be continuous: you probably want the first n derivatives to match (and thus vanish) at each endpoint.

Let’s go for a piecewise ramp function,

where $p_n(t)$ is some polynomial in t. Some counting tells you that these 2 endpoint values and 2n derivative conditions can be satisfied with a polynomial of degree 2n+1. Try changing the value of n above and see how the smoothness changes.

Now for any value of n, it’s a straightforward algebra problem to set up the polynomial and solve for the coefficients. You probably want to know the answer for the general case, and a simple approach is to do a few examples and look for the pattern. Here are the first few:

Can you spot the pattern? Don’t feel bad if you can’t, that’s why we have the OEIS. If you search for the above integers as a sequence, you’ll find A091811.

With this newfound knowledge, we can now write down the closed form for the polynomial,

But was there a better way to find this than relying on the OEIS to already have the result? But of course!

Rather than thinking about $p_n(t)$ itself, let’s think about $p_n^\prime(t)$. Since $p_n$ is strictly increasing, $p_n^\prime$ is positive, while going to zero at the endpoints. In fact it goes to zero like $t^n$ at one endpoint, and $(1-t)^n$ at the other endpoint, because we wanted n derivatives to vanish at each endpoint. Therefore we know the proportionality

The only thing to get right is the normalization, which we enforce by asking that the integral of $p_n^\prime$ is 1 at $t=1$. If you’ve spent enough time on probability and statistics, then you’ll recognize $p_n^\prime(t)$ as a special case of the Beta distribution, with shape parameters $\alpha = \beta = n+1$. So we know the normalization,

where $B(a,b)$ is the beta function, and we can now call $p_n(t)$ the regularized incomplete beta function, $p_n(t) = I_t(n+1,n+1)$.

The incomplete beta function has a representation in terms of the Gauss hypergeometric function,

The important fact for us is that we’re interested in $a=b=n+1$, in which case one of the first two arguments is a non-positive integer, and therefore the series will terminate as a finite-degree polynomial. This way, you can prove Eq. \eqref{eq:def}!