Abstract
In the paper, we presented a family M(mu, x) of approximations of the Bateman function G(x). The family M(mu, x) = G(x) for a certain whenever x is fixed and it presented asymptotical approximation of the Bateman's G-function as x -> infinity. We studied the order of convergence of the approximations M(mu, 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/e(2)-4, x) (resp.). Also, we show that the approximations M(1, x) are better than the approximation 1/x + 1/2x(2) for any mu, in an open subinterval of [1, 4/e(2)-4].