The equation appears in branching processes. s=1 is one solution but there is another solution when mu is greater than 1 which can be expressed in terms of the lambert W function

Sorry, I was careless in my last proof. I forgot the minus sign inside the lambert W function. In line 3, when i multiplied both sides by -1, I forgot to multiply the right side by -1, which results in an incorrect final answer. The actual answer is:

Sorry, I was careless in my last proof. I forgot the minus sign inside the lambert W function. In line 3, when i multiplied both sides by -1, I forgot to multiply the right side by -1, which results in an incorrect final answer. The actual answer is:

-W(-μ/e)/μ=S.