incomplete
(0.001 seconds)
1—10 of 91 matching pages
1: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►§8.17(iii) Integral Representation
… ►§8.17(iv) Recurrence Relations
… ►§8.17(v) Continued Fraction
… ►§8.17(vi) Sums
…2: 9.14 Incomplete Airy Functions
§9.14 Incomplete Airy Functions
►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …3: 8.23 Statistical Applications
§8.23 Statistical Applications
►The functions and are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …The function and its normalization play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).4: 8.24 Physical Applications
§8.24 Physical Applications
►§8.24(i) Incomplete Gamma Functions
►The function 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
…5: 8.16 Generalizations
§8.16 Generalizations
►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …6: 8.22 Mathematical Applications
§8.22 Mathematical Applications
►§8.22(i) Terminant Function
… ►§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function
… ►See Paris and Cang (1997). ►If denotes the incomplete Riemann zeta function defined by …7: 8.3 Graphics
…
►
§8.3(i) Real Variables
… ►Some monotonicity properties of and in the four quadrants of the ()-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). ► ►§8.3(ii) Complex Argument
… ►8: 8.2 Definitions and Basic Properties
§8.2 Definitions and Basic Properties
… ►The general values of the incomplete gamma functions and are defined by … ►§8.2(ii) Analytic Continuation
►In this subsection the functions and have their general values. … ►§8.2(iii) Differential Equations
…9: 8.13 Zeros
§8.13 Zeros
►§8.13(i) -Zeros of
►The function has no real zeros for . … ►Note that from (8.4.12) , . … ►§8.13(iii) -Zeros of
…10: 8.1 Special Notation
…
►Unless otherwise indicated, primes denote derivatives with respect to the argument.
►The functions treated in this chapter are the incomplete gamma functions , , , , and ; the incomplete beta functions and ; the generalized exponential integral ; the generalized sine and cosine integrals , , , and .
►Alternative notations include: Prym’s functions
, , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); , , Dingle (1973); , , Magnus et al. (1966); , , Luke (1975).