New approximations of the gamma function in terms of the digamma function

The goal of this paper is to prove the following asymptotic formula Γ ( x ) ≈ 2 π e − b ( x + b ) x exp ( − x − 1 2 ψ ( x + c ) )  as  x ∈ N , x → ∞ , where Γ is the Euler Gamma function and ψ is the digamma function, namely, the logarithmic derivative of Γ . Moreover, optimal values of parameters b , c are calculated in such a way that this asymptotic convergence is the best possible.