The main theme of the work is Wielandt's characterization of the $\Gamma$ function. Wielandt's characterization claims, that a holomorphic function $f$, defined on the right complex half-plane, for which the recursion $f(z+1) = zf(z)$ applies, is a multiple of gamma, if only f is bounded on vertical band $\{z \in \mathbb{C} |1 \le \rm {Re}(z) < 2\}$. Using this characterization, we provide alternative proofs of known results, such as the relation between Euler's functions gamma and beta, Gauss product and Euler's formula, multiplication formula of Gauss and Stirling's formula with an estimate of its error. We prove that the boundedness condition in Wielandt's characterization can be replaced by a weaker assumption, which demands, that the growth rate of function $f$ on vertical bands is not too great.
|
