The study of a function relating to stable distributions
The real function \[ t_{\alpha, \beta} (\theta) = \left( {\sin (\pi \alpha \theta + \eta_{\alpha, \beta}) \over \cos \pi \theta} \right) \left( {\cos \pi \theta \over \cos \bigl[ \pi (\alpha - 1) \theta + \eta_{\alpha, \beta} \bigr]} \right)^{(\alpha - 1)/ \alpha}, \quad - 1/2 < \theta < 1/2, \] appears in the representation of distribution functions for stable laws, see \textit{V. M. Zolotarev} [Select. Transl. Math. Stat. Probab. 6, 84-88 (1966); translation from Trudy Mat. Inst. Steklova. 71, 46-50 (1964; Zbl 0154.194)]. The values \(\eta_{\alpha, \beta}\) are certain constants. The author studies this function in detail: symmetry, zeros, continuity, monotonicity, first derivative.
scientific article; zbMATH DE number 633684