chr2017,
In the subsonic regime use the Saint Venant equations by Aiken (1947) in NACA Report 837. The report is available at:
NACA UK Mirror report description page
To the best of my knowledge calibrated airspeed has never been defined for the supersonic regime. I don't think a closed form solution can be obtained even if a definition is pursued. In the supersonic regime the pressure at the mouth of the pitot probe can be determined, for a calorically perfect gas, using the Rayleigh supersonic pitot formula. The formula accounts for the reduction in total pressure caused by the normal shock ahead of the probe, see equation 100 in NACA Report 1135 available at:
https://www.nasa.gov/sites/default/f...eport-1135.pdf Don't use the supersonic total pressure in the Saint Venant equation because it will produce a meaningless CAS value (and an erroneous TAS value).
In the hypersonic regime, say beyond about Mach 3 to 5, rotational modes become excited so the calorically perfect gas assumption becomes inappropriate and it will be necessary to work out how the adiabatic index reduces; I have a decade-old spreadsheet archived for this problem if it helps.