  We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type
over the unitary group $U(N)$ with respect to Haar measure, for finite $N$ and
including fixed external matrices. When depending only on the eigenvalues of
the unitary matrix, they can be related to a Toeplitz determinant with jump
singularities. After introducing fixed deterministic matrices as external
sources, the integrals can no longer be solved using Andr\'ei\'ef's integration
formula. Resorting to the character expansion as put forward by Balantekin, we
derive explicit determinantal formulae containing Kummer's confluent and Gauss'
hypergeometric function. They depend only on the eigenvalues of the
deterministic matrices and are analytic in the parameters of the jump
singularities. Furthermore, unitary two-matrix integrals of the same type are
proposed and solved in the same manner. When making part of the deterministic
matrices random and integrating over them, we obtain similar formulae in terms
of Pfaffian determinants. This is reminiscent to a unitary group integral found
recently by Kanazawa and Kieburg.
