I want to solve an equation involving Fermi-Dirac integrals for which I am using PolyLog functions. But I get the error:
NSolve::ifun: Inverse functions are being used by NSolve, so some solutions may not be found; use Reduce for complete solution information.
Here is the code:
ClearAll["Global`*"] (*Clear all variables*) h := 6.63*10^-34; (* J.s, Planck's constant *) hbar := (663*10^-34)/( 100*2 π); (* J.s, Reduced Planck's constant *) q := 16/10*10^-19 ; (* Coulomb, electron charge *) m0 := 91/10*10^-31; (* kg free electron rest mass *) kb := 8.61*10^-5; (*in eV/K *) gs = 2; gv = 1; mc = 0.2*m0; T = 300; nc1d = gs*gv *((2*π*mc*kb*T)/h^2)^0.5; n1d = 5*10^6; v = 1; NSolve[0.5*nc1d*(-PolyLog[1/2, -Exp[x]] - PolyLog[1/2, -Exp[x - (q*v)/(kb*T)]]) == n1d, x];
Also I want to solve $x$ for different $v$ and plot it, how do I go about it?