# term by term

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## 1—10 of 246 matching pages

##### 1: 16.26 Approximations

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►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

##### 2: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

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►Such a solution is given in terms of a Riemann theta function with two phases.
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##### 3: 19.38 Approximations

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►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m={k}^{2}$ with $$ can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸.
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►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi $ near $\pi /2$ with the improvements made in the 1970 reference.
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##### 4: 33.19 Power-Series Expansions in $r$

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33.19.1
$$f(\u03f5,\mathrm{\ell};r)={r}^{\mathrm{\ell}+1}\sum _{k=0}^{\mathrm{\infty}}{\alpha}_{k}{r}^{k},$$

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►Here $\kappa $ is defined by (33.14.6), $A(\u03f5,\mathrm{\ell})$ is defined by (33.14.11) or (33.14.12), ${\gamma}_{0}=1$, ${\gamma}_{1}=1$, and
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33.19.6
$$k(k+2\mathrm{\ell}+1){\delta}_{k}+2{\delta}_{k-1}+\u03f5{\delta}_{k-2}+2(2k+2\mathrm{\ell}+1)A(\u03f5,\mathrm{\ell}){\alpha}_{k}=0,$$
$k=2,3,\mathrm{\dots}$,

►with ${\beta}_{0}={\beta}_{1}=0$, and
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33.19.7
$${\beta}_{k}-{\beta}_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell}-2)\u03f5{\beta}_{k-2}+\frac{1}{2}(k-1)\u03f5{\gamma}_{k-2}=0,$$
$k=2,3,\mathrm{\dots}$.

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##### 5: Guide to Searching the DLMF

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term:
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►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase.
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###### Terms, Phrases and Expressions

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##### 6: 12.16 Mathematical Applications

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##### 7: 18.24 Hahn Class: Asymptotic Approximations

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►In particular, asymptotic formulas in terms of elementary functions are given when $z=x$ is real and fixed.
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►This expansion is in terms of the parabolic cylinder function and its derivative.
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►This expansion is in terms of confluent hypergeometric functions.
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►Both expansions are in terms of parabolic cylinder functions.
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###### Approximations in Terms of Laguerre Polynomials

…##### 8: 6.12 Asymptotic Expansions

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►For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$.
►For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with $p=1$.
…If the expansion is terminated at the $n$th term, then the remainder term is bounded by $1+\chi (n+1)$ times the next term.
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►The remainder terms are given by
…When $|\mathrm{ph}z|\le \frac{1}{4}\pi $, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$.
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##### 9: 7.16 Generalized Error Functions

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►These functions can be expressed in terms of the incomplete gamma function $\gamma (a,z)$ (§8.2(i)) by change of integration variable.