# with respect to integration

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## 11—20 of 36 matching pages

##### 11: Bibliography C

##### 12: 19.25 Relations to Other Functions

###### §19.25 Relations to Other Functions

… ►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. … ► … ►(${F}_{1}$ and ${F}_{D}$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.)##### 13: 4.23 Inverse Trigonometric Functions

*principal values*(or

*principal branches*) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. …The principal branches are denoted by $\mathrm{arcsin}z$, $\mathrm{arccos}z$, $\mathrm{arctan}z$, respectively. … ►are respectively …

##### 14: 3.5 Quadrature

###### §3.5(iii) Romberg Integration

►Further refinements are achieved by*Romberg integration*. … ►For these cases the integration path may need to be deformed; see §3.5(ix). … ►A second example is provided in Gil et al. (2001), where the method of contour integration is used to evaluate Scorer functions of complex argument (§9.12). … ►The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …

##### 15: 33.23 Methods of Computation

###### §33.23(iii) Integration of Defining Differential Equations

►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). …On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …##### 16: Errata

The wording was changed to make the integration variable more apparent.

The Weierstrass lattice roots ${e}_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots ${e}_{j}$, and lattice invariants ${g}_{2}$, ${g}_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

*Reported by Felix Ospald.*

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$. The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma (a,z)$. In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots}$.

*Reported 2017-07-10 by Kurt Fischer.*

These equations have been generalized to include the additional cases of $\partial {J}_{-\nu}\left(z\right)/\partial \nu $, $\partial {I}_{-\nu}\left(z\right)/\partial \nu $, respectively.

The titles have been changed to $\mu =0$ , $\nu \mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}$ , and Addendum to §14.5(ii): $\mu \mathrm{=}\mathrm{0}$, $\nu \mathrm{=}\mathrm{2}$ , respectively, in order to be more descriptive of their contents.

##### 17: 8.21 Generalized Sine and Cosine Integrals

###### §8.21(v) Special Values

… ►When $z\to \mathrm{\infty}$ with $|\mathrm{ph}z|\le \pi -\delta $ ($$), …##### 18: 1.8 Fourier Series

###### §1.8(iii) Integration and Differentiation

… ►when $f(x)$ and $g(x)$ are square-integrable and ${a}_{n},{b}_{n}$ and ${a}_{n}^{\prime},{b}_{n}^{\prime}$ are their respective Fourier coefficients. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $|{f}^{\prime \prime}(x)|$ are integrable over $(-\mathrm{\infty},\mathrm{\infty})$. … ►Suppose also that $f(x)$ is integrable on $[0,\mathrm{\infty})$ and $f(x)\to 0$ as $x\to \mathrm{\infty}$. …##### 19: 18.2 General Orthogonal Polynomials

*orthogonal on*$(a,b)$

*with respect to the weight function*$w(x)$ ($\ge 0$)

*if*…Here $w(x)$ is continuous or piecewise continuous or integrable, and such that $$ for all $n$. … ►Then a system of polynomials $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots}$, is said to be

*orthogonal*on $X$ with respect to the

*weights*${w}_{x}$ if … ►The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials ${p}_{n}(x)$ uniquely up to constant factors, which may be fixed by suitable normalization. … ►Conversely, if a system of polynomials $\{{p}_{n}(x)\}$ satisfies (18.2.10) with ${a}_{n-1}{c}_{n}>0$ ($n\ge 1$), then $\{{p}_{n}(x)\}$ is orthogonal with respect to some positive measure on $\mathbb{R}$ (Favard’s theorem). …