Frame set for shifted sinc-function

We prove that frame set ${\mathcal{F}}_g$ for imaginary shift of sinc-function $$g(t)=\frac{\sin \pi b(t-iw)}{t-iw},b,w\in \mathbb{R}\setminus \{0\}$$ can be described as ${\mathcal{F}}_g=\{(\alpha ,\beta ):\alpha \beta \le 1,\beta \le \mathrm{\mid }b\mathrm{\mid }\}\mathrm{.}$ \ In addition, we prove that ${\mathcal{F}}_g=\{(\alpha ,\beta ):\alpha \beta \le 1\}$ for window functions $g$ of the form $\frac{1}{t-iw}(1-{\sum }_{k=1}^{\mathrm{\infty }}{a}_k{e}^{2\pi i{b}_{k}t})$, such that ${\sum }_{k\ge 1}\mathrm{\mid }{a}_k\mathrm{\mid }{e}^{2\pi \mathrm{\mid }w\mathrm{\mid }{b}_k}<1$, $w{b}_k<0$.