Reexamination of Astrophysical Resonance Reaction Rate Equations for An Isolated, Narrow Resonance

The well-known astrophysical resonant reaction rate equations for an isolated narrow resonance have been reexamined. The validity of those `look reliable' assumptions used in deriving the analytic reaction rate equations has been checked, and the reality is they only hold for certain circumstances. Importantly an integration-range issue hidden in the classical integration equations has been revealed and it hints us not to use those analytic equations in the future. This work may influence all those work in which the classical narrow resonant equations were used for calculating the stellar reaction rates, especially at low stellar temperatures, and may affect the results of stellar evolution and nucleosynthesis network calculations considerably.