# §8.24 Physical Applications

## §8.24(i) Incomplete Gamma Functions

The function $\mathop{\gamma\/}\nolimits\!\left(a,x\right)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)).

## §8.24(ii) Incomplete Beta Functions

The function $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)$ appears in: Monte Carlo sampling in statistical mechanics (Kofke (2004)); analysis of packings of soft or granular objects (Prellberg and Owczarek (1995)); growth formulas in cosmology (Hamilton (2001)).

## §8.24(iii) Generalized Exponential Integral

The function $\mathop{E_{p}\/}\nolimits\!\left(x\right)$, with $p>0$, appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)).

With more general values of $p$, $\mathop{E_{p}\/}\nolimits\!\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).