$Version
(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)
Clear["Global`*"]
f = (E^(-(s/lm)) v (lm -
2 lm Hypergeometric2F1[1, -(lg/lm), 1 - lg/lm, E^(s/lg)] -
2 E^(s/lm) (s - lg Log[1 - E^(s/lg)])))/H /. {H -> 0.1, v -> 0.5,
lg -> 1, lm -> 2}
(* 5. E^(-s/2) (2 - 4 Hypergeometric2F1[-(1/2), 1, 1/2, E^s] -
2 E^(s/2) (s - Log[1 - E^s])) *)
If you are getting imaginary artifacts with your version, try using exact values and simplifying.
f2 = (E^(-(s/lm)) v (lm -
2 lm Hypergeometric2F1[1, -(lg/lm), 1 - lg/lm, E^(s/lg)] -
2 E^(s/lm) (s - lg Log[1 - E^(s/lg)])))/H /. {H -> 1/10, v -> 1/2,
lg -> 1, lm -> 2} // FunctionExpand // Simplify
(* 10 (-E^(-s/2) - s + 2 E^(-s/2) Sqrt[E^s] ArcTanh[Sqrt[E^s]] + Log[1 - E^s]) *)
Both plots are continuous
Plot[{f, f2}, {s, 3, 4},
PlotStyle -> {Automatic, Dashed}]
EDIT: Or for a more robust solution, use ComplexExpand with f2
f3 = Assuming[s > 0,
f2 // ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify]
(* 10 (-E^(-s/2) - s + 2 Log[1 + E^(s/2)]) *)
FunctionDomain[f3, s]
(* True *)
Although f3 was developed assuming s > 0, f3 == f2 for real s
Assuming[s ∈ Reals,
ComplexExpand[f2 == f3, TargetFunctions -> {Re, Im}] // FullSimplify]
(* True *)