  In this thesis a general procedure to represent the integral Bethe Ansatz
equations in the form of the Reimann-Hilbert problem is given. This allows us
to study in simple way integrable spin chains in the thermodynamic limit. Based
on the functional equations we give the procedure that allows finding the
subleading orders in the solution of various integral equations solved to the
leading order by the Wiener-Hopf technics. The integral equations are studied
in the context of the AdS/CFT correspondence, where their solution allows
verification of the integrability conjecture up to two loops of the strong
coupling expansion. In the context of the two-dimensional sigma models we
analyze the large-order behavior of the asymptotic perturbative expansion.
Obtained experience with the functional representation of the integral
equations allowed us also to solve explicitly the crossing equations that
appear in the AdS/CFT spectral problem.
