variable%20boundaries

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11: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

U. M. Ascher, R. M. M. Mattheij, and R. D. Russell (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:: Corrected reprint of the 1988 original
External Links:: ISBN 0-89871-354-4, MathReview, ZentralBlatt
Referenced by:: §3.7(iii).
Encodings:: BibTeX

…

12: Bibliography S

… ►

K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

… ►

A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

B. D. Sleeman (1966a) Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.

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Notes:: Available from the library, University of Surrey, UK
External Links:: LibraryRecord
Referenced by:: §31.9(i), §31.9(ii), Ch.31, Profile Brian D. Sleeman.
Encodings:: BibTeX

… ►

I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.

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External Links:: MathReview (R. Shail), ZentralBlatt
Referenced by:: §10.22(iv).
Encodings:: BibTeX

… ►

J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

…

13: 28.33 Physical Applications

… ►

•

Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►

§28.33(ii) Boundary-Value Problems

… ►with

W(x,y,t)=e^{\mathrm{i}\omega t}V(x,y)

, reduces to (28.32.2) with

k^{2}=\omega^{2}\rho/{\tau}

. …The boundary conditions for

\xi=\xi_{0}

(outer clamp) and

\xi=\xi_{1}

(inner clamp) yield the following equation for

q

: … ►For a visualization see Gutiérrez-Vega et al. (2003), and for references to other boundary-value problems see: …

14: 6.19 Tables

… ►

§6.19(ii) Real Variables

… ►

•

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

►

§6.19(iii) Complex Variables, $z=x+iy$

►

•

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

►

•

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

15: 11.13 Methods of Computation

… ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for

\mathbf{K}_{\nu}\left(z\right)

and

\mathbf{M}_{\nu}\left(z\right)

. … ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►For

\mathbf{M}_{\nu}\left(x\right)

both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

16: 20.13 Physical Applications

… ►For

\tau=it

, with

\alpha,t,z

real, (20.13.1) takes the form of a real-time

t

diffusion equation ►

20.13.2

\ifrac{\partial\theta}{\partial t}=\alpha\ifrac{{\partial}^{2}\theta}{{% \partial z}^{2}},

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : time, $\alpha$ : real parameter and $z$ : real variable
Referenced by:: §20.13
Permalink:: http://dlmf.nist.gov/20.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §20.13 and Ch.20

… ►

20.13.3

g(z,t)=\sqrt{\frac{\pi}{4\alpha t}}\exp\left(-\frac{z^{2}}{4\alpha t}\right)

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $t$ : time, $\alpha$ : real parameter and $z$ : real variable
Permalink:: http://dlmf.nist.gov/20.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §20.13 and Ch.20

… ►These two apparently different solutions differ only in their normalization and boundary conditions. …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). …

17: Publications

… ►

B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

… ►

B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

18: Bibliography L

… ►

R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:: ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

J. C. Light and T. Carrington Jr. (2000) Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

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External Links:: ISBN 9780470141731, Document, Link
Referenced by:: §18.38(i).
Encodings:: BibTeX

…

19: 2.11 Remainder Terms; Stokes Phenomenon

… ►In

\mathbb{C}

both the modulus and phase of the asymptotic variable

z

need to be taken into account. …Then numerical accuracy will disintegrate as the boundary rays

\operatorname{ph}z=\alpha

\operatorname{ph}z=\beta

are approached. In consequence, practical application needs to be confined to a sector

\alpha^{\prime}\leq\operatorname{ph}z\leq\beta^{\prime}

well within the sector of validity, and independent evaluations carried out on the boundaries for the smallest value of

|z|

intended to be used. … ►Since the ray

\operatorname{ph}z=\frac{3}{2}\pi

is well away from the new boundaries, the compound expansion (2.11.7) yields much more accurate results when

\operatorname{ph}z\to\frac{3}{2}\pi

. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

20: 3.7 Ordinary Differential Equations

… ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►The remaining two equations are supplied by boundary conditions of the form … ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …