  In this paper, considering the Eichler-Shimura cohomology theory for Jacobi
forms, we study connections between harmonic Maass-Jacobi forms and Jacobi
integrals. As an application we study a pairing between two Jacobi integrals,
which is defined by special values of partial $L$-functions of skew-holomorphic
Jacobi cusp forms. We obtain connections between this pairing and the Petersson
inner product for skew-holomorphic Jacobi cusp forms. This result can be
considered as analogue of Haberland formula of elliptic modular forms for
Jacobi forms.
