A Remark on Asymptotics of the Gamma Function at Infinity

The asymptotic estimate of the gamma function $|\Gamma(x+iy)| $ when $|y| $ tends to infinity is proved. The result gives a more precise one of the known formula. KEY WORDS: gamma function, asymptotic estimate MSC (2000): $33\mathrm{B}15,30\mathrm{E}15$ Let $\Gamma(z) $ be the gamma function of complex argument $z=x+iy(x,y\in \mathbb{R}=(-\infty, \infty))$ (see [1, Chapter 1]). It is known [1, 1.18(2)] the following asymptotic formula of $\Gamma(z) $ at infinity: $ ( |z|arrow \mathrm{o}\mathrm{o}, |\arg(z)|<\pi) $ , (1)$\Gamma(z)=\sqrt{2\pi}e^{-z}e^{(z-1/2)\log z}[1+O(\frac{1}{z})]$ which is called Stirling formula. As a corollary of this result, in [1, 1.18(6)] the asymptotic relation for $| $ I $(x+iy)| $ , as $|\mathrm{t}7|arrow\infty $ , is presented in the form $\lim_{|y|arrow\infty}|$ [ $(x+iy) $ $|e^{|y|\pi/2}|y|^{1/2-x}=\sqrt{2\pi} $. (2) This relation needs in correction, namely the multiplier $e^{-x} $ must be added: $\lim| $ I $(x +iy) $ $|e^{|y|\pi/2}|y|^{1[2-x}=\sqrt{2\pi}e^{-x} $. (3