Introducing LAMMbert, a concentrated liquidity AMM defined by the Lambert W function. I'll confess that just over a year ago I had never even heard of the Lambert W (LW) function. Now I seem to see people talk about it all the time. I first came across it myself whilst doing research for the DEX we're building at @eulerfinance and playing with the maths behind popular AMMs like the @Uniswap / @Bancor / @Balancer constant-product curves and @CurveFinance StableSwap curve. That's how I came to discover LAMMbert. The LW function, often denoted W(x), is a bit of a strange function, defined such that W(xe^x) = x. It is sometimes called the product log, because it is similar to the natural log function ln(x) in the sense that ln(e^x) = x. LW appears in a lot of unexpected places across many scientific disciplines. And I recently discovered it emerges naturally in the description of concentrated liquidity AMMs too! Specifically, LW appears when you're trying to build a StableSwap-like AMM that tries to blend the properties of the constant-sum (CS) and constant-product (CP) equations in order to concentrate liquidity at a particular price. I'll sketch a brief proof here. We start with a simple CS equation: x + y = 2. Multiply both sides by a concentration factor, c > 0. We have: c(x + y) = 2c. Now take a simple CP equation: xy = 1. Take the natural log of both sides, turning the left-hand side into a sum of logs. We have: ln(x) + ln(y) = 0. Both CS and CP are now defined in terms of sums. Add the modified CS and CP together: c(x + y) + ln(x) + ln(y) = 2c. Exponentiate both sides, multiply through by c, and rearrange: cye^(cy) = ce^(2c - x) / x. Take the product log W of both sides and solve for y: y = (1 / c) * W(ce^(2c - x) / x)). We call the final equation LAMMbert. Sorry. (Note: this might simplify further, but I haven't delved deeper into how W operates). Here is LAMMbert's graph in red, in between the CS and CP curves in black. It looks just like StableSwap, doesn't it? Honestly, I don't know if there's an obvious use-case for LAMMbert today. It seems a little bit difficult to work with computationally. Certainly we're not planning to use it in EulerSwap. However, I see people like @0xfoobar @optimizoor are working hard to make LW easier to compute in solidity, so maybe someone will bring it to life someday. I note that others like @guil_lambert @CK_2049 @_Dave__White_ @ad_git have all encountered LW in their own work or expressed an interest in it. So perhaps there's further relations between LAMMbert, LW, and AMMs worth exploring. Feel free to get involved if you spot something. See paper in GitHub repo below. Otherwise, here's hoping @danrobinson will add it to his AMM zoo and we can leave it in peace.