This paper presents new analytic formulas for determining the mutual inductance and the axial magnetic force between two coaxial coils in air, namely a thick circular coil with rectangular cross-section and a thin wall solenoid. The mutual inductance and the magnetic force are expressed as complete elliptical integrals of the first and second kind, Heuman's Lambda function and one well-behaved integral that must be solved numerically. All possible singular cases are automatically handled by the proposed formulas. The results of the work presented here have been verified by the filament method and previously published data. The new formulas provide a substantially simple alternative over previously published approaches, which involve either numerical techniques (finite element method, boundary element method, method of moments) or other semianalytic or analytic approaches.