  We propose formulas of the nuclear beta-decay rate that are useful in a
practical calculation. The decay rate is determined by the product of the
lepton and hadron current densities. A widely used formula relies upon the fact
that the low-energy lepton wave functions in a nucleus can be well approximated
by a constant and linear to the radius for the $s$-wave and $p$-wave wave
functions, respectively. We find, however, the deviation from such a simple
approximation is evident for heavy nuclei with large $Z$ by numerically solving
the Dirac equation. In our proposed formulas, the neutrino wave function is
treated exactly as a plane wave, while the electron wave function is obtained
by iteratively solving the integral equation, thus we can control the
uncertainty of the approximate wave function. The leading-order approximation
gives a formula equivalent to the conventional one and overestimates the decay
rate. We demonstrate that the next-to-leading-order formula reproduces well the
exact result for a schematic transition density as well as a microscopic one
obtained by a nuclear energy-density functional method.
