TITLE: Is there an analytic approximation to the minimum function?
QUESTION [17 upvotes]: I am looking for an analytic function that approximates the minimum function. i.e.,
$|f(x_1,x_2) - \min(x_1,x_2)| < \zeta$ for some $\zeta$ that may be related to $|x_1 - x_2|$. Or may be a series $f_1,f_2,....$, where $\lim_{n \to \infty} \zeta = 0$.

REPLY [11 votes]: The only function that I know has this property is essentially
$$\max(x, y) \sim \frac{x e^{kx} + y e^{ky}}{e^{kx} + e^{ky}}$$
for large $k$.
(@wnoise was close to the answer. In this case there is no problem if $x=y$.)
This works for positive or negative numbers and tends to the maximum for $k\to\infty$. Other formulas based on powers are only valid for positive numbers (as $\sqrt[n]{x^n + y^n}$).
The integral counterpart is
$$ \frac{\int{f(x) e^{kf(x)} dx}}{\int{e^{kf(x)} dx}} $$
There is a small caveat if you want to implement this numerically you need to have an estimate of the maximum in the first place (and integrate over shifted values), otherwise the the integral (or even the sum) is going to under/overflow pretty easily.
This is related to the Laplace's method. Note that I asked a related question:
Real approximation to the maximum using Laplace's method integral