1 + z = γ (1 − β cos θ)
θ = 0: source moving along the photon (approaching). θ = π: receding. Spheres expand at c from each emission event. The observed z uses retarded time — where the source was when this light was emitted, which is not where it is now. A fly-by at closest approach still looks blueshifted.
receding: 1 + z = √((1+β)/(1−β))