# term by term

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## 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 $0\leq m<1$ 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 $\phi$ near $\pi/2$ with the improvements made in the 1970 reference. …
##### 4: 33.19 Power-Series Expansions in $r$
Here $\kappa$ is defined by (33.14.6), $A(\epsilon,\ell)$ is defined by (33.14.11) or (33.14.12), $\gamma_{0}=1$, $\gamma_{1}=1$, and …
33.19.6 $k(k+2\ell+1)\delta_{k}+2\delta_{k-1}+\epsilon\delta_{k-2}+2(2k+2\ell+1)A(% \epsilon,\ell)\alpha_{k}=0,$ $k=2,3,\dots$,
with $\beta_{0}=\beta_{1}=0$, and
33.19.7 $\beta_{k}-\beta_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\beta_{k-2}+\tfrac{1% }{2}(k-1)\epsilon\gamma_{k-2}=0,$ $k=2,3,\dots$.
##### 5: Guide to Searching the DLMF
###### Terms, Phrases and Expressions
Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators:
• term:

a textual word, a number, or a math symbol.

• 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. …
• Single-letter terms

##### 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. …
##### 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 $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi$, 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 $\operatorname{ph}z=0$. …
##### 9: 7.16 Generalized Error Functions
These functions can be expressed in terms of the incomplete gamma function $\gamma\left(a,z\right)$8.2(i)) by change of integration variable.
##### 10: 16.7 Relations to Other Functions
Further representations of special functions in terms of ${{}_{p}F_{q}}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${{}_{q+1}F_{q}}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).