Complete asymptotic expansions for the relativistic Fermi-Dirac integral

Asymptotic Expansions of Integral Mean of Polygamma Functions

Let s,t be two given real numbers, s = t and m ∈ N . We determine the coefficients aj(s,t) in the asymptotic expansion of integral (or differential) mean of polygamma functions ψ (m)(x) : 1 t− s ∫ t s ψ (m)(x+u)du ∼ ψ (m) ( x ∞ ∑ j=0 aj(s,t) x j ) , x → ∞. We derive the recursive relations for polynomials aj(t,s) , and also as polynomials in intrinsic variables α = 2 (s + t − 1) , β = 4 [1− (t ...

Asymptotic Expansions for Integrals

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales.

We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral, [Formula: see text] as [Formula: see text], using the distributional approach of McClure & Wong. We find that, for half-integer orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the ...

Asymptotic Expansions

Asymptotic expansions of functions are useful in statistics in three main ways. Firstly, conventional asymptotic expansions of special functions are useful for approximate computation of integrals arising in statistical calculations. An example given below is the use of Stirling's approximation to the gamma function. Second, asymptotic expansions of density or distribution functions of estimato...

Fermi-Dirac Statistics

Fermi-Dirac statistics are one of two kinds of statistics exhibited by!identical quantum particles, the other being !Bose-Einstein statistics. Such particles are called fermions and bosons respectively (the terminology is due to Dirac [1902-1984] [1]). In the light of the !spin-statistics theorem, and consistent with observation, fermions are invariably spinors (of half-integral spin), whilst b...