# Transcendental equation analytical solving

I want to solve a transcendental equation analytically, the equation basically is

Solve[-2.50663 gs gv ((kb mc T)/h^2)^0.5 PolyLog[1/2, -E^y] ==
1.25331 gs gv ((kb mc T)/
h^2)^0.5 (-PolyLog[1/2, -E^x] - PolyLog[1/2, -E^(-vd + x)]), y]


I want to solve for y and get it in terms of the other variables. I can't plot and find the solution since I am looking for analytical answer. What is the best way to do so?

• Reduce is sometimes more powerful than Solve for non-polynomial problems. If we assume a few things about your problem then your problem "simplifies" down to Reduce[PolyLog[1/2,-E^y]==c,y] for a complicated expression c, but it replies "This system cannot be solved with the methods available to Reduce". – Bill Apr 28 '19 at 17:00
• @Bill then how should I go about next? – Indeterminate Apr 28 '19 at 17:33
• Look into the literature on PolyLog and see if there are any known inverses. Since Plot[PolyLog[1/2,-E^y],{y,-10,10}] looks nice and simple and Real then could you possibly express this as an infinite series and get a solution that way? Ask very bright and experienced mathematicians if they are aware of any ways to get analytical solutions to your problem. I know of nothing else. – Bill Apr 28 '19 at 19:25