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Similarly to the standard effective range expansion that is done near the threshold energy, we obtain a generalized power-series expansion of the multi-channel Jost-matrix that can be done near an arbitrary point on the Riemann surface of the energy within the domain of its analyticity. In order to do this, we analytically factorize its momentum dependencies at all the branching points on the Riemann surface. The remaining single-valued matrix functions of the energy are then expanded in thedoi:10.1088/1751-8113/44/11/115303 fatcat:hpsk2p57snemtjrpkzyw26ndmm