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1: 1.14 Integral Transforms
Assume that f ( t ) is piecewise continuous on [ 0 , ) and of exponential growth, that is, constants M and α exist such that …
2: 8.24 Physical Applications
§8.24 Physical Applications
The function I x ( a , b ) 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 E p ( x ) , with p > 0 , appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)). With more general values of p , E p ( x ) 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)).
3: 15.19 Methods of Computation
However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. …
4: 13.19 Asymptotic Expansions for Large Argument
13.19.1 M κ , μ ( x ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 x x - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! x - s , μ - κ - 1 2 , - 3 2 , .
13.19.2 M κ , μ ( z ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 z z - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! z - s + Γ ( 1 + 2 μ ) Γ ( 1 2 + μ + κ ) e - 1 2 z ± ( 1 2 + μ - κ ) π i z κ s = 0 ( 1 2 + μ - κ ) s ( 1 2 - μ - κ ) s s ! ( - z ) - s , - 1 2 π + δ ± ph z 3 2 π - δ ,
For an asymptotic expansion of W κ , μ ( z ) as z that is valid in the sector | ph z | π - δ and where the real parameters κ , μ are subject to the growth conditions κ = o ( z ) , μ = o ( z ) , see Wong (1973a). …
5: 28.29 Definitions and Basic Properties
iff e π i ν is an eigenvalue of the matrix …
28.29.10 F ν ( z ) = e i ν z P ν ( z ) ,
A nontrivial solution w ( z ) is either a Floquet solution with respect to ν , or w ( z + π ) - e i ν π w ( z ) is a Floquet solution with respect to - ν . … Its order of growth for | λ | is exactly 1 2 ; see Magnus and Winkler (1966, Chapter II, pp. 19–28). …
6: Bibliography V
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.
  • P. Verbeeck (1970) Rational approximations for exponential integrals E n ( x ) . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.
  • H. Volkmer (1998) On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.