We present new relations for integrals of complex gamma functions. We show that starting from properties of the elliptic hypergeometric integrals, and using some limiting procedures, one can get vast number of identities for integrals of products of the hyperbolic and complex gamma functions.
The latter integrals have physical interpretation of partition functions for different types of quantum field theories dualities in various dimensions, and the corresponding limiting procedures can be viewed as dimensional reduction and multiplet decoupling.