#### LESSONS OF EASTER ISLAND

March 17th, 2016

Considerable effort has been invested in deriving approximate expressions for Fermi-Dirac integrals as reviewed elsewhere (Blakemore 1982).

One compact approximation for negative arguments is based on the following expression:

(C.13)

where the constant Cj is chosen to minimise error over the desired range. This expression has a similar expansion to that of the integrals of interest with an identical first term. The second term is also identical if Cj is given the value 2-(j + ]). Cj chosen in the range:

2-(j+1) <> C± <>±(1II±(0)-1)

will give the best results for negative arguments, where the value on the right is ensures that the expression gives the correct value for n = 0. The value of the left

hand side will cause the expression to underestimate the integral while the value on the right will always overestimate it.

The expression is accurate over all orders with j > 0 to better than about 1% accuracy, and considerably better for n < -1. For the inverse problem, that of

calculating n given /±, the expression leads to:

П = ln( 1 /I ± + C j) (C.15)

which is accurate to better than 0.01 kT for n negative.

For negative arguments, the integrals can be evaluated to any predetermined level of accuracy by using only a finite number of terms of the series expansion of Eq. (C.5). By bounding the error in the remaining terms, it follows that:

r±, , m-1 (+1)r+1 exp(rn) ,± Ij(n) =^, rJ+1 +A |
(C.16) |

A± (+1 )m+1 exp(mn) mj+1[1 ± [m/(m + 1)]j+1 exp(n)] |
(C.17) |

Re lative Error < A± exp( n) |
(C.18) |

Values of m of unity, so that only the A term is evaluated, are accurate to much better than 1% accuracy for n < -4. More terms are required for higher n values. Working through the series until the magnitude of the last term evaluated is consistent with the desired relative error and dividing this by the factor in square brackets before adding to the accumulated sum would be an appropriate procedure.

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