# right-hand

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## 1—10 of 109 matching pages

##### 1: 25.19 Tables

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Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots}$, 20D (p. 811); ${\mathrm{Li}}_{2}\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ}({1}^{\circ}){30}^{\circ}({2}^{\circ}){90}^{\circ}({5}^{\circ}){180}^{\circ}$, $f(\theta )+\theta \mathrm{ln}\theta $, $\theta =0({1}^{\circ}){15}^{\circ}$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

##### 2: 32.15 Orthogonal Polynomials

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►For this result and applications see Fokas et al. (1991): in this reference, on the right-hand side of Eq.
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##### 3: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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##### 4: 16.5 Integral Representations and Integrals

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►In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when $$.
In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector $$; compare §16.2(iii).
Lastly, when $p>q+1$ the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side.
In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta $ is an arbitrary small positive constant.
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##### 5: 24.19 Methods of Computation

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►If ${\stackrel{~}{N}}_{2n}$ denotes the right-hand side of (24.19.1) but with the second product taken only for $p\le \lfloor {(\pi \mathrm{e})}^{-1}2n\rfloor +1$, then ${N}_{2n}=\lceil {\stackrel{~}{N}}_{2n}\rceil $ for $n\ge 2$.
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##### 6: 19.21 Connection Formulas

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19.21.7
$$(x-y){R}_{D}(y,z,x)+(z-y){R}_{D}(x,y,z)=3{R}_{F}(x,y,z)-3{y}^{1/2}{x}^{-1/2}{z}^{-1/2},$$

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19.21.8
$${R}_{D}(y,z,x)+{R}_{D}(z,x,y)+{R}_{D}(x,y,z)=3{x}^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

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19.21.10
$$2{R}_{G}(x,y,z)=z{R}_{F}(x,y,z)-\frac{1}{3}(x-z)(y-z){R}_{D}(x,y,z)+{x}^{1/2}{y}^{1/2}{z}^{-1/2},$$
$z\ne 0$.

►Because ${R}_{G}$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when ${R}_{G}$ is calculated from ${R}_{F}$ and ${R}_{D}$ (see §19.36(i)).
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##### 7: 19.25 Relations to Other Functions

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19.25.7
$$E(\varphi ,k)=2{R}_{G}(c-1,c-{k}^{2},c)-(c-1){R}_{F}(c-1,c-{k}^{2},c)-\sqrt{c-1}\sqrt{c-{k}^{2}}/\sqrt{c},$$

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►All terms on the right-hand sides are nonnegative when ${k}^{2}\le 0$, $0\le {k}^{2}\le 1$, or $1\le {k}^{2}\le c$, respectively.
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►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14).
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►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which $$, for some $j$.
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►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which $$, for some $j$.
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##### 8: 16.10 Expansions in Series of ${}_{p}{}^{}F_{q}^{}$ Functions

##### 9: 1.6 Vectors and Vector-Valued Functions

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►where $\mathbf{n}$ is the unit vector normal to $\mathbf{a}$ and $\mathbf{b}$ whose direction is determined by the right-hand rule; see Figure 1.6.1.
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