# representation by squares

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## 1—10 of 17 matching pages

##### 1: 27.13 Functions

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###### §27.13(iv) Representation by Squares

…##### 2: 22.16 Related Functions

##### 3: Bibliography G

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Representations of Integers as Sums of Squares.
Springer-Verlag, New York.
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##### 4: Bibliography E

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On the representations of a number as a sum of three squares.
Proc. London Math. Soc. (3) 9, pp. 575–594.
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##### 5: 1.17 Integral and Series Representations of the Dirac Delta

###### §1.17 Integral and Series Representations of the Dirac Delta

… ►###### §1.17(ii) Integral Representations

… ►Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation … ►###### Coulomb Functions (§33.14(iv))

… ►###### §1.17(iii) Series Representations

…##### 6: Bibliography M

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On the representation of numbers as a sum of $2r$
squares.
Quarterly Journal of Math. 48, pp. 93–104.
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##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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###### §1.18(ii) ${{L}}^{{2}}$ spaces on intervals in ${\mathbb{R}}$

►Let ${X}{=}{[}{a}{,}{b}{]}$ or ${[}{a}{,}{b}{)}$ or ${(}{a}{,}{b}{]}$ or ${(}{a}{,}{b}{)}$ be a (possibly infinite, or semi-infinite) interval in ${\mathbb{R}}$ . For a Lebesgue–Stieltjes measure ${d}{\alpha}$ on ${X}$ let ${{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on ${X}$ which are square integrable with respect to ${d}{\alpha}$ , …Functions ${f}{,}{g}{\in}{{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ for which ${\u27e8}{f}{-}{g}{,}{f}{-}{g}{\u27e9}{=}{0}$ are identified with each other. The space ${{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ becomes a separable Hilbert space with inner product … ►Assume that ${{\left\{}{{\varphi}}_{{n}}{\right\}}}_{{n}{=}{0}}^{{\mathrm{\infty}}}$ is an orthonormal basis of ${{L}}^{{2}}{}{\left(}{X}{\right)}$ . The formulas in §1.18(i) are then: … ►for ${f}{}{(}{x}{)}{\in}{{L}}^{{2}}$ and piece-wise continuous, with convergence as discussed in §1.18(ii). …##### 8: Bibliography

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Integral representation of Kelvin functions and their derivatives with respect to the order.
Z. Angew. Math. Phys. 42 (5), pp. 708–714.
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On magic squares constructed by the uniform step method.
Proc. Amer. Math. Soc. 2 (4), pp. 557–565.
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Integral representations for Jacobi polynomials and some applications.
J. Math. Anal. Appl. 26 (2), pp. 411–437.
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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum.
SIAM J. Math. Anal. 5, pp. 119–124.
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##### 9: 26.15 Permutations: Matrix Notation

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*sign of the permutation*$\sigma $ is the sign of the determinant of its matrix representation. The*inversion number*of $\sigma $ is a sum of products of pairs of entries in the matrix representation of $\sigma $: … ►Let ${r}_{j}(B)$ be the number of ways of placing $j$ nonattacking rooks on the squares of $B$. …##### 10: Errata

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Section 16.11(i)
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Chapter 19
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Additions
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Equation (11.11.1)
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Equation (35.7.3)
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Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

*Reported by Luc Maisonobe on 2021-06-07*

Pochhammer symbol representations for the functions ${F}_{k}(\nu )$ and ${G}_{k}(\nu )$ were inserted.

Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_{2}{}^{}F_{1}^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.