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Here's a short routine for computing the Mach number from the Prandtl-Meyer equation:
prandtlMeyerMachNumber[γ_?InexactNumberQ, ν_?InexactNumberQ] :=
Module[{prec = Precision[{γ, ν}], β, η, λ, m},
λ = Sqrt[SetPrecision[(γ - 1)/(γ + 1), ∞]];
η = ν + π (1 - 1/λ)/2;
m = ((3 + γ) η^2/12 + 2/(1 - γ))/η;
Sqrt[β^2 + ...
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