Reference books on Lambert function

Question

Does anyone have a good reference book on the Lambert W function? I need it as reference for my thesis particularly, as reference for this

The general solution to $$x=a+be^{cx}$$ is $$x=a-\frac{1}{c}W(-bce^{ac})$$

I found this on Wikipedia but it does not link the actual reference. Thanks for you help!

Answer 1

There's a book by I. Mező from 2022 that might work for you. It deals with the equation $x+b^x=a$ in section 1.1.3, and from that you should be able to derive the result you quoted.

Answer 2

You can also provide a proof yourself in your thesis. It is easy: \begin{align*} x &= a + b{\rm e}^{cx} \\[1ex] x - a &= b{\rm e}^{cx} \\[1ex] (x - a){\rm e}^{ - cx} &= b \\[1ex] (x - a){\rm e}^{ - cx + ac} & = b{\rm e}^{ac} \\[1ex] (x - a){\rm e}^{ - c(x - a)} & = b{\rm e}^{ac} \\[1ex] - c(x - a){\rm e}^{ - c(x - a)} & = - bc{\rm e}^{ac} \\[1ex] - c(x - a) & = W( - bc{\rm e}^{ac} ) \\[1ex] x & = a - \frac{1}{c}W( - bc{\rm e}^{ac} ). \end{align*}