with respect to summation
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1: 18.3 Definitions
2: 3.11 Approximation Techniques
3: 25.11 Hurwitz Zeta Function
4: 1.6 Vectors and VectorValued 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 ${\mathbf{\Phi}}_{1}$ and ${\mathbf{\Phi}}_{2}$ 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
6: 23.18 Modular Transformations
7: 1.14 Integral Transforms
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 $t\to \mathrm{\infty}$, where $\delta >0$. …8: Errata
The following additions were made in Chapter 1:

Section 1.2
New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

Section 1.3
The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
 Section 1.4

Section 1.8
In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

Section 1.10
New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

Section 1.13
New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: SturmLiouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

Section 1.14(i)
Another form of Parseval’s formula, (1.14.7_5).

Section 1.16
We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

Section 1.17
Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

Section 1.18
An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).
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 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.
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 20170128 by Richard Paris.
Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.
Reported 20101107 by Layne Watson.