About the Project

exponential%20growth

AdvancedHelp

(0.004 seconds)

1—10 of 469 matching pages

1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(ii) Graphics
§8.19(v) Recurrence Relation and Derivatives
§8.19(ix) Inequalities
§8.19(x) Integrals
2: 4.2 Definitions
log e x = ln x is also called the Napierian or hyperbolic logarithm. …
§4.2(iii) The Exponential Function
The function exp is an entire function of z , with no real or complex zeros. …
4.2.32 e z = exp z ,
but the general value of e z is …
3: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
The principal value of the exponential integral E 1 ( z ) is defined by … Ein ( z ) is sometimes called the complementary exponential integral. … The logarithmic integral is defined by …
4: 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)).
5: 6.19 Tables
  • Abramowitz and Stegun (1964, Chapter 5) includes x 1 Si ( x ) , x 2 Cin ( x ) , x 1 Ein ( x ) , x 1 Ein ( x ) , x = 0 ( .01 ) 0.5 ; Si ( x ) , Ci ( x ) , Ei ( x ) , E 1 ( x ) , x = 0.5 ( .01 ) 2 ; Si ( x ) , Ci ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x = 2 ( .1 ) 10 ; x f ( x ) , x 2 g ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x 1 = 0 ( .005 ) 0.1 ; Si ( π x ) , Cin ( π x ) , x = 0 ( .1 ) 10 . Accuracy varies but is within the range 8S–11S.

  • Zhang and Jin (1996, pp. 652, 689) includes Si ( x ) , Ci ( x ) , x = 0 ( .5 ) 20 ( 2 ) 30 , 8D; Ei ( x ) , E 1 ( x ) , x = [ 0 , 100 ] , 8S.

  • Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of z e z E 1 ( z ) , x = 19 ( 1 ) 20 , y = 0 ( 1 ) 20 , 6D; e z E 1 ( z ) , x = 4 ( .5 ) 2 , y = 0 ( .2 ) 1 , 6D; E 1 ( z ) + ln z , x = 2 ( .5 ) 2.5 , y = 0 ( .2 ) 1 , 6D.

  • Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of E 1 ( z ) , ± x = 0.5 , 1 , 3 , 5 , 10 , 15 , 20 , 50 , 100 , y = 0 ( .5 ) 1 ( 1 ) 5 ( 5 ) 30 , 50 , 100 , 8S.

  • 6: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • §8.26(iv) Generalized Exponential Integral
  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 7: 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.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • 8: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • Clenshaw (1962) gives Chebyshev coefficients for E 1 ( x ) ln | x | for 4 x 4 and e x E 1 ( x ) for x 4 (20D).

  • Luke (1969b, pp. 411–414) gives rational approximations for Ein ( z ) .

  • 9: 15.19 Methods of Computation
    For z it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when z = e ± π i / 3 . This is because the linear transformations map the pair { e π i / 3 , e π i / 3 } onto itself. However, by appropriate choice of the constant z 0 in (15.15.1) we can obtain an infinite series that converges on a disk containing z = e ± π i / 3 . … 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. …
    10: 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 ,
    13.19.3 W κ , μ ( z ) e 1 2 z z κ s = 0 ( 1 2 + μ κ ) s ( 1 2 μ κ ) s s ! ( z ) s , | 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). …