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1: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(iii) Integral Representation

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§8.17(iv) Recurrence Relations

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§8.17(v) Continued Fraction

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§8.17(vi) Sums

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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 $P(a,x)$ and $Q(a,x)$ are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ 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 $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).

4: 8.24 Physical Applications

§8.24 Physical Applications

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§8.24(i) Incomplete Gamma Functions

►The function $\gamma (a,x)$ 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

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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

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§8.22(i) Terminant Function

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

… ►See Paris and Cang (1997). ►If ${\zeta }_x(s)$ denotes the incomplete Riemann zeta function defined by …

7: 8.3 Graphics

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§8.3(i) Real Variables

… ►Some monotonicity properties of ${\gamma }^{*}(a,x)$ and $\mathrm{\Gamma }(a,x)$ in the four quadrants of the ($a,x$)-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). ►

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Figure 8.3.7: $x^{-a}-{\gamma }^{*}(a,x)$ (= $x^{-a}Q(a,x)$), $0\le x\le 4$, $-5\le a\le 5$. Magnify 3D Help

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§8.3(ii) Complex Argument

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Figure 8.3.16: ${\gamma }^{*}(2.5,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. Magnify 3D Help

8: 8.2 Definitions and Basic Properties

§8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by … ►

§8.2(ii) Analytic Continuation

►In this subsection the functions $\gamma$ and $\mathrm{\Gamma }$ have their general values. … ►

§8.2(iii) Differential Equations

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9: 8.13 Zeros

§8.13 Zeros

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§8.13(i) $x$-Zeros of ${\gamma }^{*}(a,x)$

►The function ${\gamma }^{*}(a,x)$ has no real zeros for $a\ge 0$. … ►Note that from (8.4.12) ${\gamma }^{*}(-n,0)=0$, $n=1,2,3,\mathrm{\dots }$. … ►

§8.13(iii) $a$-Zeros of ${\gamma }^{*}(a,x)$

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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 $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{*}(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).