Find Function with r^2 Near Zero & Approaching 0

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SUMMARY

The discussion focuses on finding a mathematical function that behaves like \( r^2 \) near zero and approaches zero rapidly as \( r \) tends to infinity. Participants suggest multiplying two functions: one that behaves like \( r^2 \) near zero and another that decays faster than \( r^2 \) increases. Recommended functions include \( \frac{r^2}{1 + r^3} \) and \( r^2 e^{-r} \) as viable options to meet these criteria.

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I'm looking for a function which has a behavior r^2 near zero and approaching fast to zero when r tends to infinity. Thanks for any hints.
 
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Multiply two functions together where one behaves like ##r^2## near zero (how about ##r^2##) and the other behaves like 1 near zero and decays to zero more quickly than ##r^2## increases. After you get the basic second function you can manipulated it to make the product of the two functions fit other constraints you have.
 
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You want to pick the "second function" to make life easy for whatever you are doing. There are plenty of options, like
$$\frac{r^2}{1 + r^3}, \quad r^2 e^{-r}, \quad \text{etc.}$$
 
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