# Rational approximation of Heaviside function

1. Nov 15, 2013

### hilbert2

Hi, could someone please help me with this one: I'd need to form a sequence of rational functions $R_{n}(x)$ such that $\lim_{n \to \infty} R_{n}(x)=\theta(x)$, where $\theta(x)$ is the Heaviside step function. The functions $R_{n}(x)$ should preferably be limited in range, i.e. for some real number $M$, $|R_{n}(x)|<M$ for all $n$ and $x$. This is not a homework problem, I just happen to need a rational approximation for the step function.

2. Nov 16, 2013

### pwsnafu

The problem is that rational functions can only have one horizontal asymptote, but Heaviside has two. So you need to be more specific by what you want.

3. Nov 16, 2013

### hilbert2

Ok, thanks for the answer. I was looking for a step function approximation for which the inverse Laplace transform can be calculated in closed form. I probably have to approach the problem some other way.