  We study the production of light nuclei in the coalescence mechanism in Au-Au
collisions at midrapidity at $\sqrt{s_{NN}}=3$ GeV. We derive analytic formulas
of momentum distributions of two bodies, three bodies and four nucleons
coalescing into light nuclei, respectively. We naturally explain the transverse
momentum spectra of the deuteron ($d$), triton ($t$), helium-3 ($^3$He) and
helium-4 ($^4$He). We reproduce the data of yield rapidity densities and
averaged transverse momenta of $d$, $t$, $^3$He and $^4$He. We give proportions
of contributions from different coalescence sources for $t$, $^3$He and $^4$He
in their productions. We find that besides nucleon coalescence,
nucleon$+$nucleus coalescence and nucleus$+$nucleus coalescence may play
requisite roles in light nuclei production in Au-Au collisions at
$\sqrt{s_{NN}}=3$ GeV.
