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1: 20.14 Methods of Computation

… ►For instance, the first three terms of (20.2.1) give the value of ${\theta }_1\left(2-\mathrm{i}\right|\mathrm{i})$ ($={\theta }_1(2-\mathrm{i},{\mathrm{e}}^{-\pi })$) to 12 decimal places. ►For values of $\left|q\right|$ near $1$ the transformations of §20.7(viii) can be used to replace $\tau$ with a value that has a larger imaginary part and hence a smaller value of $\left|q\right|$. …In theory, starting from any value of $\tau$, a finite number of applications of the transformations $\tau \to \tau +1$ and $\tau \to -1/\tau$ will result in a value of $\tau$ with $\mathrm{\Im }\tau \ge \sqrt{3}/2$; see §23.18. …

2: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(i) Fourier Transform

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§1.14(iii) Laplace Transform

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Fourier Transform

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Laplace Transform

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3: 16.4 Argument Unity

… ►There are two types of three-term identities for ${}_3{}^{}F_2^{}$’s. … ►The other three-term relations are extensions of Kummer’s relations for ${}_2{}^{}F_1^{}$’s given in §15.10(ii). … ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. … ►One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … ►Relations between three solutions of three-term recurrence relations are given by Masson (1991). …

4: 6.13 Zeros

… ►For these results, together with the next three terms in (6.13.2), see MacLeod (2002a). …

5: Bibliography G

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Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§10.74(iv), §14.32, §3.10(iii), §3.6(iii).

Encodings:

BibTeX

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Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Algorithm 490: The Dilogarithm function of a real argument [S22]. Comm. ACM 18 (4), pp. 200–202.

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Notes:

Double-precision Fortran, minimum accuracy 15D. For remark see ACM Trans. Math. Software 2 (1976), p. 112

External Links:

ISSN 0001-0782, Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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6: Bibliography D

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Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96080-5, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

Encodings:

BibTeX

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

BibTeX

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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7: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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Heine’s First Transformation

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Heine’s Third Transformation

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Fine’s Second Transformation

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Fine’s Third Transformation

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Three-Term ${}_2{}^{}\varphi _1^{}$ Transformations

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8: 18.33 Polynomials Orthogonal on the Unit Circle

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. … ►while combination of (18.33.27) and (18.33.23) gives the three-term recurrence relation …

9: 2.11 Remainder Terms; Stokes Phenomenon

… ►Taking $m=10$ in (2.11.2), the first three terms give us the approximation … ►

§2.11(vi) Direct Numerical Transformations

… ► … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. ►However, direct numerical transformations need to be used with care. …

10: 4.13 Lambert $W$-Function

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4.13.10 $$W_k\left(z\right)\sim {\xi }_k-\ln {\xi }_k+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{{\xi }_k^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln {\xi }_k\right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).

Referenced by:

§4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first threeterms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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4.13.11 $$W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim -\eta -\ln \eta +\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\eta }^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln \eta \right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$

Notes:

See Corless et al. (1996, p. 350).

Referenced by:

§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first threeterms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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