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31: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►For numerical values of $d_{k,n}$ to 30D for $k=0(1)9$ and $n=0(1)N_k$, where $N_k=28-4\lfloor k/2\rfloor$, see DiDonato and Morris (1986). … ►Higher coefficients $A_k(\chi )$, $B_k(\chi )$, up to $k=8$, are given in Paris (2002b). ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). … ►

Inverse Function

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32: 4.13 Lambert $W$-Function

… ►where $t\ge 0$ for $W_0$, $t\le 0$ for $W_{\pm 1}$ on the relevant branch cuts, …See Jeffrey and Murdoch (2017) for an explicit representation for the $c_n$ in terms of associated Stirling numbers. … ►For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. … ►For the foregoing results and further information see Borwein and Corless (1999), Corless et al. (1996), de Bruijn (1961, pp. 25–28), Olver (1997b, pp. 12–13), and Siewert and Burniston (1973). ►For a generalization of the Lambert $W$-function connected to the three-body problem see Scott et al. (2006), Scott et al. (2013) and Scott et al. (2014).

33: 3.4 Differentiation

… ►The $B_k^n$ are the differentiated Lagrangian interpolation coefficients: … ►where $c_n$ is defined by (3.3.12), with numerical values as in §3.3(ii). … ►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. Taking $C$ to be a circle of radius $r$ centered at $x_0$, we obtain … ►With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. …

34: 27.2 Functions

… ►They tend to thin out among the large integers, but this thinning out is not completely regular. … ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►and if $\varphi \left(n\right)$ is the smallest positive integer $f$ such that $a^f\equiv 1(modn)$, then $a$ is a primitive root mod $n$. …Such a set is a reduced residue system modulo $n$. … ►where $p^a$ is a prime power with $a\ge 1$; otherwise $\mathrm{\Lambda }\left(n\right)=0$. …