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.
Eq. 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.
Eq. 2 |
We can hammer equation 1 into the form of Equation 2 by the process shown in Equation 3.
Eq. 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.