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1: 27.10 Periodic Number-Theoretic Functions

… ►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. … ► $G(n,\chi )$ is separable for some $n$ if … ►For any Dirichlet character $\chi (modk)$, $G(n,\chi )$ is separable for $n$ if $\left(n,k\right)=1$, and is separable for every $n$ if and only if $G(n,\chi )=0$ whenever $\left(n,k\right)>1$. For a primitive character $\chi (modk)$, $G(n,\chi )$ is separable for every $n$, and … ►Conversely, if $G(n,\chi )$ is separable for every $n$, then $\chi$ is primitive (mod $k$). …

2: 20 Theta Functions

Chapter 20 Theta Functions

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3: Bibliography K

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The group $\mathrm{O}(4)$, separation of variables and the hydrogen atom. SIAM J. Appl. Math. 30 (4), pp. 630–664.

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External Links:

Document, MathReview (Anthony J. Bracken), ZentralBlatt

Referenced by:

§31.17(i).

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Separation of Variables for Riemannian Spaces of Constant Curvature. Longman Scientific & Technical, Harlow.

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External Links:

ISBN 0-582-98807-1, MathReview (H. Kilp), ZentralBlatt

Referenced by:

§31.17(i).

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

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4: 15.20 Software

… ►References to research software that is available in other ways is listed separately. …

5: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.

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External Links:

Document, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§12.13(ii), §12.17.

Encodings:

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Symmetry and Separation of Variables. Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam.

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Notes:

Encyclopedia of Mathematics and its Applications, Vol. 4

External Links:

ISBN 0-201-13503-5, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§31.17(i).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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6: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►On separation of variables into cylindrical coordinates, the Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$ and $K_n\left(x\right)$, all appear. … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …

7: 26.10 Integer Partitions: Other Restrictions

… ►If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. … ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …

8: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

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A separating surface for the Painlevé differential equation $x^{\prime \prime }=x^2-t$ . J. Math. Anal. Appl. 193 (3), pp. 817–831.

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External Links:

ISSN 0022-247X, Document, MathReview (W. S. Loud), ZentralBlatt

Referenced by:

§32.17.

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9: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes $p\le x$: … ►Gauss and Legendre conjectured that $\pi \left(x\right)$ is asymptotic to $x/\ln x$ as $x\to \mathrm{\infty }$: …(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. …

10: 31.10 Integral Equations and Representations

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Kernel Functions

… ►where $\sigma$ is a separation constant. … ►

31.10.10 $$𝒦(z,t)={(zt-a)}^{\frac{1}{2}-\delta -\sigma }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}-\delta -\sigma +\alpha ,\frac{1}{2}-\delta -\sigma +\beta }{\gamma };\frac{zt}{a}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}+\delta +\sigma ,-\frac{1}{2}+ϵ-\sigma }{\delta };\frac{a(z-1)(t-1)}{(a-1)(zt-a)}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel and $\sigma$: separation constant

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http://dlmf.nist.gov/31.10.E10

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See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

… ►where ${\sigma }_1$ and ${\sigma }_2$ are separation constants. … ►and ${\sigma }_1$ and ${\sigma }_2$ are separation constants. …