Some approximations of the Bateman's G--function.

In the paper, we presented a family M(μ, x) of approximations of the Bateman function G(x). The family M(μ, x) = G(x) for a certain μ whenever x is fixed and it presented asymptotical approximation of the Bateman's G-function as x → ∞. We studied the or- der of convergence of the approximations M(μ, x) of the function G(x). Some properties and bounds of the error are deduced. We presented new sharp double inequality of G(x) with the upper and lower bounds M(1; x) and M( 4/e2-4, x) (resp.). Also, we show that the approximations M(μ, x) are better than the approximation 1/x + 12x2 for any μ in an open subinterval of [1; 4/e2-4]. [ABSTRACT FROM AUTHOR]