Precision Models

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Correction Factors

Rayleigh correction factor

\[c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}\] To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula:

EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))

Gaussian correction factor

\[\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})\] The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:

=EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))

Bessel correction factor

\[c_{B}(n) = \frac{n}{n-1}\]