Alternatives to the n = 1 Hermite function

Question

I'm modeling some data that appear to follow an n = 1 Hermite function.

Here's the equation:

Here's the graph:

Are there any other functions that have the same shape that I should check out? The function should be odd, asymptotic at y = 0 as x approaches ∞ and -∞, and it should also be confined to quadrants I and III.

Thanks!

Answer 1

Let us drop the front coefficient and consider an approximation of

$$f(x)=xe^{-x^2/2}$$

Here is one with an excellent degree of approximation on interval $(-1.8,1.8)$:

$$g(x)=\frac{x}{1+\frac12x^2+\frac18x^4+\tfrac{1}{32}x^6}$$

obtained by "tuning" a Taylor expansion of $f$.

The absolute value of the max. error between the two curves of $f$ and $g$ is bounded by $0.005$.

This approximation could be slightly improved by taking a certain supplementary term in $x^8$ in the denominator.