How to approximate Heaviside function by polynomial

Question

I have a Heaviside smooth function that defined as $$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$

I want to use polynominal to approximate the Heaviside function. Could you suggest to me a solution? Thanks

UPDATE: This is Bombyx mori result in blue line and my expected result is red line

Answer 1

Here is a solution: $$ \frac{1}{2}+\frac{1}{\pi}(\frac{x}{\epsilon}-\frac{x^3}{3\epsilon^{3}}+\frac{x^{5}}{5\epsilon^5}\cdots) $$

Answer 2

Althogh I think Bombyx mori's solution is great and simple, maybe the following approximation is better for you on the interval $[-1,1]$. $$ p_\varepsilon(x) = -\tfrac{1}{6\pi}\left(16\arctan\left(\tfrac{1}{2\varepsilon}\right)-8\arctan\left(\tfrac{1}{\varepsilon}\right)\right)x^3 - \tfrac{1}{6\pi}\left(2\arctan\left(\tfrac{1}{\varepsilon}\right)-16\arctan\left(\tfrac{1}{2\varepsilon}\right)\right)x+\tfrac12. $$ I've got this by making a polynomial interpolation for the data points: $$(-1,H_\varepsilon(-1)), (-1/2,H_\varepsilon(-1/2)), (0,H_\varepsilon(0)),(1/2,H_\varepsilon(1/2)), (1,H_\varepsilon(1)).$$