DLMF: Untitled Document

… ►

J. A. Gaunt (1929) The triplets of helium. Philos. Trans. Roy. Soc. London Ser. A 228, pp. 151–196.

ⓘ

External Links:: Document, ZentralBlatt (Möglich, Dr. F.)
Referenced by:: §34.3(vii).
Encodings:: BibTeX

… ►

A. Gil and J. Segura (2001) DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.

ⓘ

External Links:: Document, Code, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

W. Gröbner and N. Hofreiter (1949) Integraltafel. Erster Teil. Unbestimmte Integrale. Springer-Verlag, Vienna.

ⓘ

Notes:: Reprintings with corrections appeared in 1961 and 1965
External Links:: MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §1.4(iv), §23.14, §4.10(iii), §4.26(v), §4.40(v).
Encodings:: BibTeX

… ►

H. Gupta (1935) A table of partitions. Proc. London Math. Soc. (2) 39, pp. 142–149.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §27.21.
Encodings:: BibTeX

►

H. Gupta (1937) A table of partitions (II). Proc. London Math. Soc. (2) 42, pp. 546–549.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §27.21.
Encodings:: BibTeX

…

… ►

M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

ⓘ

External Links:: MathReview (J. G. van der Corput), ZentralBlatt
Referenced by:: §30.9(iii).
Encodings:: BibTeX

… ►

C. L. Adler, J. A. Lock, B. R. Stone, and C. J. Garcia (1997) High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder. J. Opt. Soc. Amer. A 14 (6), pp. 1305–1315.

ⓘ

External Links:: FullText
Referenced by:: §36.14(ii).
Encodings:: BibTeX

… ►

G. B. Airy (1838) On the intensity of light in the neighbourhood of a caustic. Trans. Camb. Phil. Soc. 6, pp. 379–402.

ⓘ

Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

►

G. B. Airy (1849) Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

ⓘ

Referenced by:: §9.16.
Encodings:: BibTeX

… ►

D. E. Amos (1985) A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.

ⓘ

External Links:: FullText
Referenced by:: 1st item, B. Döring (1966).
Encodings:: BibTeX

…

… ►The asymptotic behavior of

\lambda^{m}_{n}\left(\gamma^{2}\right)

and

a^{m}_{n,k}(\gamma^{2})

as

n\to\infty

in descending powers of

2n+1

is derived in Meixner (1944). The cases of large

m

, and of large

m

and large

|\gamma|

, are studied in Abramowitz (1949). …

… ►

4.26.7

\int e^{ax}\sin\left(bx\right)\,\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\sin% \left(bx\right)-b\cos\left(bx\right)),

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable
A&S Ref:: 4.3.136
Permalink:: http://dlmf.nist.gov/4.26.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.26(ii), §4.26 and Ch.4

►

4.26.8

\int e^{ax}\cos\left(bx\right)\,\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\cos% \left(bx\right)+b\sin\left(bx\right)).

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable
A&S Ref:: 4.3.137
Permalink:: http://dlmf.nist.gov/4.26.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.26(ii), §4.26 and Ch.4

… ►

4.26.9

\int_{0}^{\pi}\sin\left(mt\right)\sin\left(nt\right)\,\mathrm{d}t=0,

m\neq n

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer and $n$ : integer
A&S Ref:: 4.3.140
Permalink:: http://dlmf.nist.gov/4.26.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

►

4.26.10

\int_{0}^{\pi}\cos\left(mt\right)\cos\left(nt\right)\,\mathrm{d}t=0,

m\neq n

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $n$ : integer
A&S Ref:: 4.3.140
Permalink:: http://dlmf.nist.gov/4.26.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

… ►(Here and elsewhere in this subsection

\delta

is a constant such that

0<\delta<\pi

.) … ►

A(r,\theta)

is a harmonic function in polar coordinates (1.9.27), and … ►The lower limit

0

of the integral in (1.15.47) can be replaced by any constant

a\leq x

. Also, we can replace the lower and upper limits of the integral by

x

and

a

, respectively. … ►and either

\left|a_{n}\right|\leq K

or

a_{n}\geq 0

, then …

… ►where

a_{m}

is the

m

th negative zero of

\operatorname{Ai}\left(x\right)

(§9.9(i)). … ►

18.16.15

x_{n,m}<\nu+2^{\frac{2}{3}}a_{m}\nu^{\frac{1}{3}}+2^{-\frac{2}{3}}{a_{m}}^{2}% \nu^{-\frac{1}{3}},

ⓘ

Symbols:: $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\nu$ and $x$ : real variable
Referenced by:: §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.16.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

… ►

18.16.17

x_{n,m}=(2n+1)^{\frac{1}{2}}+2^{-\frac{1}{3}}(2n+1)^{-\frac{1}{6}}a_{m}+% \epsilon_{n,m},

ⓘ

Symbols:: $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $\epsilon_{n,m}$ and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.16.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §18.16(v), §18.16 and Ch.18

►where

a_{m}

is the

m

th negative zero of

\operatorname{Ai}\left(x\right)

(§9.9(i)),

\epsilon_{n,m}<0

, and as

n\to\infty

with

m

fixed … ►In the notation of this reference

x_{n,m}=u_{a,m}

,

\mu=\sqrt{2n+1}

, and

\alpha=\mu^{-\frac{4}{3}}a_{m}

. …

… ►For extensive tables of integrals, see Apelblat (1983), Bierens de Haan (1867), Gradshteyn and Ryzhik (2000), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b). … ►A function

f(x)

is square-integrable if … ►With

a<b

, the total variation of

f(x)

on a finite or infinite interval

(a,b)

is … ►A function

f(x)

is convex on

(a,b)

if …A similar definition applies to closed intervals

[a,b]

. …

… ►

T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

ⓘ

External Links:: ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(v).
Encodings:: BibTeX

… ►

J. Meixner (1944) Die Laméschen Wellenfunktionen des Drehellipsoids. Forschungsbericht No. 1952 ZWB (German).

ⓘ

Notes:: English translation: Lamé wave functions of the ellipsoid of revolution appeared as NACA Technical Memorandum No. 1224, 1949
External Links:: MathReview
Referenced by:: §30.9(iii).
Encodings:: BibTeX

… ►

A. Michaeli (1996) Asymptotic analysis of edge-excited currents on a convex face of a perfectly conducting wedge under overlapping penumbra region conditions. IEEE Trans. Antennas and Propagation 44 (1), pp. 97–101.

ⓘ

External Links:: Document
Referenced by:: §9.14.
Encodings:: BibTeX

… ►

MuPAD (commercial interactive system and Matlab toolbox) SciFace Software, Paderborn, Germany.

ⓘ

Notes:: Computer algebra system for doing symbolic and exact algebraic computations as well as numerical calculations with almost arbitrary accuracy
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

L. A. Muraveĭ (1976) Zeros of the function

Ai^{\prime}(z)-\sigma Ai(z)

. Differential Equations 11, pp. 797–811.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §9.9(i).
Encodings:: BibTeX

…

… ►These results are derived formally in Sips (1949, 1959, 1965). … ►Let

x=\tfrac{1}{2}\pi-\mu h^{-\ifrac{1}{4}}

, where

\mu

is a constant such that

\mu\geq 1

, and

s=2m+1

. … ►The approximations apply when the parameters

a

and

q

are real and large, and are uniform with respect to various regions in the

z

-plane. … ►These approximations apply when

q

and

a

are real and

q\to\infty

. They are uniform with respect to

a

when

-2q\leq a\leq(2-\delta)q

, where

\delta

is an arbitrary constant such that

0<\delta<4

, and also with respect to

z

in the semi-infinite strip given by

0\leq\Re z\leq\pi

and

\Im z\geq 0

. …

… ►

J. Segura (1998) A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi).
Encodings:: BibTeX

… ►

J. Shao and P. Hänggi (1998) Decoherent dynamics of a two-level system coupled to a sea of spins. Phys. Rev. Lett. 81 (26), pp. 5710–5713.

ⓘ

External Links:: Document
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

A. Sidi (2003) Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-66159-5, MathReview (Thomas A. Atchison), ZentralBlatt
Referenced by:: §3.9(vi).
Encodings:: BibTeX

… ►

R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).

ⓘ

External Links:: FullText, MathReview (M. Strutt), ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

… ►

A. N. Stokes (1980) A stable quotient-difference algorithm. Math. Comp. 34 (150), pp. 515–519.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §3.10(ii).
Encodings:: BibTeX

…

艾里森山大学商业管理文凭证书【somewhat微aptao168】1949a

11—20 of 889 matching pages

11: Bibliography G

12: Bibliography

13: 30.9 Asymptotic Approximations and Expansions

14: 4.26 Integrals

15: 1.15 Summability Methods

16: 18.16 Zeros

17: 1.4 Calculus of One Variable

18: Bibliography M

19: 28.8 Asymptotic Expansions for Large $q$

20: Bibliography S