representation by squares

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1: 27.13 Functions

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§27.13(iv) Representation by Squares

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2: 22.16 Related Functions

… ►See Figure 22.16.2. …

3: Bibliography G

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Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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External Links:

ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt

Referenced by:

§27.13(iv).

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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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External Links:

Document, MathReview (S. Chowla), ZentralBlatt

Referenced by:

§20.12(i).

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BibTeX

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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

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§1.17(ii) Integral Representations

… ►Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation … ►

Sine and Cosine Functions

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§1.17(iii) Series Representations

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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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External Links:

ZentralBlatt

Referenced by:

§27.13(iv).

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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 $ℝ$

… ►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 $d\alpha$, …The space $L^2(X,d\alpha )$ becomes a separable Hilbert space with inner product … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►The well must be deep and broad enough to allow existence of such $L^2$ discrete states. …

8: 18.39 Applications in the Physical Sciences

… ►where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►with an infinite set of orthonormal $L^2$ eigenfunctions … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. …Here tridiagonal representations of simple Schrödinger operators play a similar role. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

9: 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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ISSN 0044-2275, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.61(v), §10.64.

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BibTeX

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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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ISSN 0002-9939, Fulltext, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.17.

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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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External Links:

Document, MathReview (D. L. Colton), ZentralBlatt

Referenced by:

§18.17(iv).

Encodings:

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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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ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

Encodings:

BibTeX

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10: 26.15 Permutations: Matrix Notation

… ►The 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 $B$. …