Rational approximation of Heaviside function

[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.

 

[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.

 

[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.