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21: 3.4 Differentiation

… ►If

f^{(n+2)}(x)

is continuous on the interval

I

defined in §3.3(i), then the remainder in (3.4.1) is given by … ►where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain … ►

First-Order

… ►For additional formulas involving values of

\nabla^{2}u

and

\nabla^{4}u

on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

►

•

Curtis (1964a) tabulates $P_{\ell}(\epsilon,r)$ , $Q_{\ell}(\epsilon,r)$ (§33.1), and related functions for $\ell=0,1,2$ and $\epsilon=-2(.2)2$ , with $x=0(.1)4$ for $\epsilon<0$ and $x=0(.1)10$ for $\epsilon\geq 0$ ; 6D.

…

23: Errata

… ►

Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

… ►

Subsection 1.16(viii)

An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

… ►

Section 17.9

The title was changed from Transformations of Higher ${{}_{r}\phi_{r}}$ Functions to Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions.

… ►

Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

…

24: Staff

… ►

William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

►

Gerhard Wolf, University of Duisberg-Essen, Chap. 28

… ►

William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

…

25: 23 Weierstrass Elliptic and Modular
Functions

…

26: Bibliography B

… ►

G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

… ►

A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

… ►

W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

… ►

S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

…

27: Bibliography P

… ►

R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

ⓘ

External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

… ►

R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

… ►

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

ⓘ

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).
Encodings:: BibTeX

…

28: Bibliography G

… ►

B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

… ►

A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

ⓘ

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

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

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

… ►

Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

ⓘ

External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

…

29: 36 Integrals with Coalescing Saddles

…

30: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …