# with respect to summation

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## 1—10 of 17 matching pages

##### 1: 18.3 Definitions
In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_{n}\left(x\right)$, $n=0,1,\dots,N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\dots,N+1$, of $T_{N+1}\left(x\right)$: …
##### 2: 3.11 Approximation Techniques
When $n>0$ and $0\leq j\leq n$, $0\leq k\leq n$, … Now suppose that $X_{k\ell}=0$ when $k\neq\ell$, that is, the functions $\phi_{k}(x)$ are orthogonal with respect to weighted summation on the discrete set $x_{1},x_{2},\dots,x_{J}$. … …
##### 3: 25.11 Hurwitz Zeta Function
The Riemann zeta function is a special case: … For other series expansions similar to (25.11.10) see Coffey (2008). … In (25.11.18)–(25.11.24) primes on $\zeta$ denote derivatives with respect to $s$. … When $a=1$, (25.11.35) reduces to (25.2.3). … uniformly with respect to bounded nonnegative values of $\alpha$. …
##### 4: 1.6 Vectors and Vector-Valued Functions
###### Einstein Summation Convention
Much vector algebra involves summation over suffices of products of vector components. In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. … If $\boldsymbol{{\Phi}}_{1}$ and $\boldsymbol{{\Phi}}_{2}$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_{1}$ and $D_{2}$ respectively, then …
##### 5: 1.17 Integral and Series Representations of the Dirac Delta
subject to certain conditions on the function $\phi(x)$. … The last condition is satisfied, for example, when $\phi(x)=O\left({\mathrm{e}}^{\alpha x^{2}}\right)$ as $x\to\pm\infty$, where $\alpha$ is a real constant. … Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9) $\&$ (1.14.11), (1.14.10) $\&$ (1.14.12), respectively. … In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^{2}$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …
##### 6: 23.18 Modular Transformations
according as the elements $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ of $\mathcal{A}$ in (23.15.3) have the respective forms …Here e and o are generic symbols for even and odd integers, respectively. …
23.18.6 $\varepsilon(\mathcal{A})=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)% \right),$
23.18.7 ${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$ $c>0$.
##### 7: 1.14 Integral Transforms
If $f(t)$ is absolutely integrable on $(-\infty,\infty)$, then $F(x)$ is continuous, $F(x)\to 0$ as $x\to\pm\infty$, and …
###### Poisson’s Summation Formula
The Fourier cosine transform and Fourier sine transform are defined respectively by … If also $\lim_{t\to 0+}f(t)/t$ exists, then … Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta}\right)$ as $t\to\infty$, where $\delta>0$. …
##### 8: Errata

• Subsection 19.25(vi)

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.

• Subsections 14.5(ii), 14.5(vi)

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

• Equation (8.12.18)
8.12.18 $\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\Gamma\left(a% \right)}{\left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum% _{k=1}^{\infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)}$

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

• Equation (26.7.6)
26.7.6 $B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}B\left(k\right)$

Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.

Reported 2010-11-07 by Layne Watson.

• ##### 9: 8.12 Uniform Asymptotic Expansions for Large Parameter
in each case uniformly with respect to $\lambda$ in the sector $|\operatorname{ph}\lambda|\leq 2\pi-\delta$ ($<2\pi$). … For the asymptotic behavior of $c_{k}(\eta)$ as $k\to\infty$ see Dunster et al. (1998) and Olde Daalhuis (1998c). … Higher coefficients $A_{k}(\chi)$, $B_{k}(\chi)$, up to $k=8$, are given in Paris (2002b). … For asymptotic expansions, as $a\to\infty$, of the inverse function $x=x(a,q)$ that satisfies the equation …These expansions involve the inverse error function $\operatorname{inverfc}\left(x\right)$7.17), and are uniform with respect to $q\in[0,1]$. …
##### 10: 11.10 Anger–Weber Functions
11.10.5 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=f(\nu,z),$
###### §11.10(vi) Relations to Other Functions
11.10.27 $\left.\frac{\partial}{\partial\nu}\mathbf{J}_{\nu}\left(z\right)\right|_{\nu=0% }=\tfrac{1}{2}\pi\mathbf{H}_{0}\left(z\right),$
where the prime on the second summation symbols means that the first term is to be halved. …