The Endless Beta Integrals

We consider a special degeneration limit ${\omega }_1\to -{\omega }_2$ (or $b\to {\rm i}$inthecontextof$2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is discovered in the limit ${\omega }_1\to {\omega }_2$ (or $b\to 1$).