We clarify the structure of the hadamard regularization scheme. The Hadamard elementary function is obtained in the Hadamard series expansion for a generally coupled massive scalar field in a general curved spacetime. We obtain the constraint equations for the state-dependent terms in the series. The expectation value of the stress energy tensor is regularized with the point-splitting method. Consequently we clarify the state dependence of the expectation value of the stress energy tensor. It obeys the conservation law by virtue of the constraint equations. They reduce to a simple expression for a conformally coupled massless scalar field in the Schwarzschild geometry. We give a simple solution of the equation which leads to the stress energy tensor that behaves isotropically in the asymptotic region and is regular on the past and future horizons.