in terms of elementary functions

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11: 18.15 Asymptotic Approximations

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In Terms of Elementary Functions

… ►For more powerful asymptotic expansions as $n\to \mathrm{\infty }$ in terms of elementary functions that apply uniformly when , $-1+\delta \le t\le 1-\delta$, or , where $t=x/\sqrt{2n+1}$ and $\delta$ is again an arbitrary small positive constant, see §§12.10(i)–12.10(iv) and 12.10(vi). …

12: 32.2 Differential Equations

… ►However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …

13: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►These approximations are in terms of elementary functions. …

14: 2.8 Differential Equations with a Parameter

… ►Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that depend on arbitrarily chosen reference points ${\alpha }_j$, are $C^{\mathrm{\infty }}$ or analytic on $𝚫$, and as $u\to \mathrm{\infty }$ …

15: 7.7 Integral Representations

… ►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. …

16: 2.11 Remainder Terms; Stokes Phenomenon

… ►Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector . …

17: 30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …

18: 14.15 Uniform Asymptotic Approximations

… ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed. …

19: 19.15 Advantages of Symmetry

… ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). …

20: 6.7 Integral Representations

… ►Many integrals with exponentials and rational functions, for example, integrals of the type $\int {\mathrm{e}}^{z}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be represented in finite form in terms of the function $E_1\left(z\right)$ and elementary functions; see Lebedev (1965, p. 42). …