An integral formula for Taylor coefficients of a class of analytic functions
Let \(f(z)\) be a function analytic in \(\mathbb{C}\setminus[1,\infty)\) and let \(\Delta f(t) \lim_{\varepsilon\to 0^+} (f(t+i\varepsilon) -f(t-i \varepsilon))\). In this paper the following result is proved: Theorem. Let \(f\) be analytic in \(\mathbb{C}\setminus[1,\infty)\) and \(a_n\) be its Taylor coefficients. If \(f\) is bounded along \((1,\infty)\) such that \(| f(z)|= \circ (| z|^k)\) for some \(k\) and that \(\lim_{z\to 1}(1-z)f(z)=0\) and if \(\Delta f(x)\) exists for every \(x>1\), then for \(a_n\) the following integral representation \(a_n={1\over 2\Pi i} \int^\infty_1 {\Delta f(t) \over t^{n+1}} dt\) holds. Some applications of this theorem are also presented.
scientific article; zbMATH DE number 1289961