The article is devoted to the study of the fifth Painlev'e equation which has 4 complex parameters α, β, γ, δ. By methods of Power Geometry we look for asymptotic expansions of solutions to the equation when x → ∞. Ten power expansions with two exponential additions each are obtained when α≠0. Six of them are over integer powers x (they have been already known) and four are over half-integer powers (they are new).  When α=0  we computed 4 one-parameter families of exponential asymptotic forms   y(x) and 3 one-parameter families of complicated expansions x=x(y). All exponential additions, exponential asymptotic forms and complicated expansions have not been known before. Here we improve the method of computation of exponential additions to the power expansions as well.

Mathematical problems and theory of numerical methods