In mathematics, the quantum dilogarithm is a special function defined by the formula It is the same as the q-exponential function
be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation
Faddeev's quantum dilogarithm
is defined by the following formula: where the contour of integration
goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin.
The same function can be described by the integral formula of Woronowicz: Ludvig Faddeev discovered the quantum pentagon identity: where
are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation and the inversion relation The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.