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31: 2.11 Remainder Terms; Stokes Phenomenon

… ►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited. … ►In particular, … ►

§2.11(vi) Direct Numerical Transformations

… ►shows that this direct estimate is correct to almost 3D. … ►However, direct numerical transformations need to be used with care. …

32: Notices

… ►Questions regarding this copyright policy should be directed to dlmf-feedback. … ►Certain products, commercial and otherwise, are mentioned in the DLMF. … ►If you feel you have found an error in DLMF, please see Possible Errors in DLMF. … ►Using special functions in applications often requires computing them. …We do this in two ways. …

33: 9.17 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. …On the remaining rays, given by $\mathrm{ph}z=\pm \frac{1}{3}\pi$ and $\pi$, integration can proceed in either direction. … ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. …In these cases boundary-value methods need to be used instead; see §3.7(iii). … ►This method was used in the computation of the tables in §9.9(v). …

34: Bibliography L

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The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

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§31.12.

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

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Guide to Tables in the Theory of Numbers. Bulletin of the National Research Council, No. 105, National Research Council, Washington, D.C..

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Notes:

Reprinted in 1961.

External Links:

MathReview (R. D. Carmichael), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21, §27.21.

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Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

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Referenced by:

§3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

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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).

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35: How to Cite

… ►For example, the sentence in the LaTeX source: …(assuming \usepackage{hyperref} in the preamble, and NIST:DLMF being the key for DLMF) … ►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. … ►For convenience, the permalink can be found in the pop-up ‘Info box’ associated with each item in the site.

36: 15.6 Integral Representations

… ►In addition: ►In (15.6.1) all functions in the integrand assume their principal values. … ►It encircles $t=0$ and $t=1$ once in the positive direction, and then once in the negative direction. …However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by $-1$. … ►In each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values. …

37: 29.20 Methods of Computation

… ►The eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^m\left(k^2\right)$, and the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►The numerical computations described in Jansen (1977) are based in part upon this method. … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … ►Alternatively, the zeros can be found by locating the maximum of function $g$ in (29.12.11).

38: 1.10 Functions of a Complex Variable

… ►In addition, … ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. Similarly if $D=ℂ\setminus [0,\mathrm{\infty })$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. … ►If the path circles a branch point at $z=a$, $k$ times in the positive sense, and returns to $z_0$ without encircling any other branch point, then its value is denoted conventionally as $F((z_0-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$. … ►Many properties are a direct consequence of this representation: Taking the $x$-derivative gives us …

39: Bibliography P

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A First Course in Fluid Dynamics. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-25416-7; 0-521-27424-9, MathReview, ZentralBlatt

Referenced by:

§18.39(v).

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Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

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External Links:

MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

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Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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2nd item.

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A First Course in Real Analysis. 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97437-7, MathReview (Gerald J. Janusz)

Referenced by:

§1.5(iv), §1.8(i).

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Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:

Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.

External Links:

ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).

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40: 1.9 Calculus of a Complex Variable

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Real and Imaginary Parts

… ► … ► … ►If $f(z)=u(x,y)+\mathrm{i}v(x,y)$ is analytic in an open domain $D$, then $u$ and $v$ are harmonic in $D$, that is, … ►The angle between $C_1$ and $C_2$ at $z_0$ is the angle between the tangents to the two arcs at $z_0$, that is, the difference of the signed angles that the tangents make with the positive direction of the real axis. …