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 |  |
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 , definition of W function |
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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.
Figure 1: Numerical Solution in Mathcad.