# generalized integrals

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##### 2: 8.21 Generalized Sine and Cosine Integrals
###### §8.21 Generalized Sine and Cosine Integrals
When $\operatorname{ph}z=0$ (and when $a\neq-1,-3,-5,\dots$, in the case of $\operatorname{Si}\left(a,z\right)$, or $a\neq 0,-2,-4,\dots$, in the case of $\operatorname{Ci}\left(a,z\right)$) the principal values of $\operatorname{si}\left(a,z\right)$, $\operatorname{ci}\left(a,z\right)$, $\operatorname{Si}\left(a,z\right)$, and $\operatorname{Ci}\left(a,z\right)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … From here on it is assumed that unless indicated otherwise the functions $\operatorname{si}\left(a,z\right)$, $\operatorname{ci}\left(a,z\right)$, $\operatorname{Si}\left(a,z\right)$, and $\operatorname{Ci}\left(a,z\right)$ have their principal values. … For the corresponding expansions for $\operatorname{si}\left(a,z\right)$ and $\operatorname{ci}\left(a,z\right)$ apply (8.21.20) and (8.21.21).
##### 4: 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 $\gamma\left(a,z\right)$, $\Gamma\left(a,z\right)$, $\gamma^{*}\left(a,z\right)$, $P\left(a,z\right)$, and $Q\left(a,z\right)$; the incomplete beta functions $\mathrm{B}_{x}\left(a,b\right)$ and $I_{x}\left(a,b\right)$; the generalized exponential integral $E_{p}\left(z\right)$; the generalized sine and cosine integrals $\operatorname{si}\left(a,z\right)$, $\operatorname{ci}\left(a,z\right)$, $\operatorname{Si}\left(a,z\right)$, and $\operatorname{Ci}\left(a,z\right)$. Alternative notations include: Prym’s functions $P_{z}(a)=\gamma\left(a,z\right)$, $Q_{z}(a)=\Gamma\left(a,z\right)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma\left(a+1,z\right)$, $[a,z]!=\Gamma\left(a+1,z\right)$, Dingle (1973); $B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)$, $I(a,b,x)=I_{x}\left(a,b\right)$, Magnus et al. (1966); $\operatorname{Si}\left(a,x\right)\to\operatorname{Si}\left(1-a,x\right)$, $\operatorname{Ci}\left(a,x\right)\to\operatorname{Ci}\left(1-a,x\right)$, Luke (1975).
##### 5: 9.13 Generalized Airy Functions
Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: …
$B_{0}\left(z,0\right)=0$ ,
Each of the functions $A_{k}\left(z,p\right)$ and $B_{k}\left(z,p\right)$ satisfies the differential equation … The $A_{k}\left(z,p\right)$ are related by …
##### 6: 19.35 Other Applications
###### §19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …
##### 7: 7.16 Generalized Error Functions
###### §7.16 Generalized Error Functions
Generalizations of the error function and Dawson’s integral are $\int_{0}^{x}e^{-t^{p}}\,\mathrm{d}t$ and $\int_{0}^{x}e^{t^{p}}\,\mathrm{d}t$. …
##### 8: 8.24 Physical Applications
###### §8.24(iii) Generalized Exponential Integral
With more general values of $p$, $E_{p}\left(x\right)$ 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)).
##### 9: 16.24 Physical Applications
###### §16.24(i) Random Walks
Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. …
##### 10: 8.27 Approximations
###### §8.27(ii) Generalized Exponential Integral
• Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$.