# term by term

(0.002 seconds)

## 1—10 of 247 matching pages

##### 1: 16.26 Approximations

…
►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)

…
►Such a solution is given in terms of a Riemann theta function with two phases.
…

##### 3: 19.38 Approximations

…
►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.
…
►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.
…

##### 4: 33.19 Power-Series Expansions in $r$

…
►

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

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

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

…
►
term:
…
►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.
…
►
•
…

###### Terms, Phrases and Expressions

►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: ►a textual word, a number, or a math symbol.

Single-letter terms

##### 6: 12.16 Mathematical Applications

…
►

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

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

###### Approximations in Terms of Laguerre Polynomials

…##### 8: 6.12 Asymptotic Expansions

…
►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.
…
►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 $\mathrm{ph}z=0$.
…

##### 9: 7.16 Generalized Error Functions

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