Evaluation of weak measurements to all orders

Some exact formulas of the expectation values and probability densities in a weak measurement for an operator A that satisfies the property A(2) = 1 are derived. These formulas include all-order effects of the unitary evolution due to the von Neumann interaction. These are valid not only in the weak-measurement regime, but also in the strong-measurement regime and tell us the connection between these two regimes. Using these formulas, arguments of the optimization of the signal amplification and the signal-to-noise ratio are developed in two typical experimental setups.