A quick math problem came by my cube today that was worth sharing. I was asked if there was a closed-form solution to Equation 1.
This problem does not have a closed form solution using the “everyday” functions, but it is solvable using Lambert’s W function. Recall that Lambert’s W function has the property shown in Equation 2.
We can hammer equation 1 into the form of Equation 2 by the process shown in Equation 3.
|, definition of W function|
We now have a closed form solution, but it requires the use of a relatively uncommon function. Let’s compute the numeric solution. As shown in Figure 1, I drop into Mathcad for this part.