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11: Bibliography T

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Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.

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External Links:

Document

Referenced by:

§31.17(ii).

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Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

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Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

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12: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. … ►Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

13: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). … ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …

14: 10.73 Physical Applications

… ► … ► … ► … ►

§10.73(ii) Spherical Bessel Functions

… ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …

15: Bibliography J

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A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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External Links:

ISSN 0020-0255, Document, MathReview, ZentralBlatt

Referenced by:

§12.16.

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Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (3), pp. 407–448.

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External Links:

ISSN 0167-2789, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.4(i), §32.4(ii), §32.4(iv), §32.4(v), §32.6(ii), §32.6(iii), §32.6(iv), §32.6(v), §32.6(v), §32.6(v).

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

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Notes:

Second edition of Jones and Sleeman (2003)

External Links:

ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt

Referenced by:

Profile Brian D. Sleeman.

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

Encodings:

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On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§32.12(i).

Encodings:

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16: 18.12 Generating Functions

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18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n,$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.8.12))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2)

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.2_5 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\alpha +1};\frac{1-R-z}{2}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\beta +1};\frac{1-R+z}{2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\gamma \right)}_n{\left(\alpha +\beta +1-\gamma \right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ $R=\sqrt{1-2xz+z^2}$, ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Rainville (1960, §141, (2)); Brafman’s generating function(proved)

Source:

Koekoek et al. (2010, (9.8.15))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E2_5

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3_5 $$\frac{1+z}{{(1-2xz+z^2)}^{\beta +\frac{3}{2}}}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\beta +2\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\beta +1,\beta )}\left(x\right)z^n,$$ ,

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute $\alpha =\beta +1$ in (18.12.3) and use (15.4.6).

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E3_5

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.17 $$\frac{1+2xz+4z^2}{{\left(1+4z^2\right)}^{\frac{3}{2}}}\exp \left(\frac{4x^2{z}^2}{1+4z^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{H_n\left(x\right)}{\lfloor n/2\rfloor !}{z}^n,$$ .

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\exp z$: exponential function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.15.14))

Referenced by:

§18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E17

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

17: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

18: Bibliography S

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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External Links:

ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:

Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

Encodings:

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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.

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ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

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A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 567–577.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

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The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§1.13(v).

Encodings:

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19: Bibliography M

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

Encodings:

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

Encodings:

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Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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External Links:

ISSN 0025-5831, Document, MathReview, ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

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Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.10(ii).

Encodings:

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20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►where the integral kernel is given by …Equation (1.18.19) is often called the completeness relation. … ►

Hermite’s Differential Equation, $X=(-\mathrm{\infty },\mathrm{\infty })$

… ►Applying equations (1.18.29) and (1.18.30), the complete set of normalized eigenfunctions being … ► See §18.39 for discussion of Schrödinger equations and operators. …