DLMF: Untitled Document

… ►Let

h(t)={J_{\nu}}^{2}\left(t\right)

and

f(t)=1/(1+t)

. …In the half-plane

\Re z>\max(0,-2\nu)

, the product

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)\mathscr{M}\mskip-3.0muh% \mskip 3.0mu\left(z\right)

has a pole of order two at each positive integer, and … ►We now apply (2.5.5) with

\max(0,-2\nu)<c<1

, and then translate the integration contour to the right. … ►From (2.5.26) and (2.5.28), it follows that both

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)

and

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)

are defined in the half-plane

\Re z>\max(1-b,-c)

. … ►Put

x=1/\zeta

and break the integration range at

t=1

, as in (2.5.23) and (2.5.24). …

… ►

1.7.2

\sum^{n}_{j=1}a_{j}b_{j}\leq\left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(% \sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}.

ⓘ

Symbols:: $j$ : integer, $n$ : nonnegative integer, $q$ : number and $p>1$
A&S Ref:: 3.2.8
Permalink:: http://dlmf.nist.gov/1.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(i), §1.7(i), §1.7 and Ch.1

… ►Conversely, if

\sum^{n}_{j=1}a_{j}b_{j}\leq A^{1/p}B^{1/q}

for all

b_{j}

such that

\sum^{n}_{j=1}b_{j}^{q}\leq B

, then

\sum^{n}_{j=1}a_{j}^{p}\leq A

. … ►

1.7.3

\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p}\leq\left(\sum^{n}_{j=1}a_{j% }^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}.

ⓘ

Symbols:: $j$ : integer, $n$ : nonnegative integer and $p>1$
A&S Ref:: 3.2.12
Permalink:: http://dlmf.nist.gov/1.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(i), §1.7(i), §1.7 and Ch.1

… ►

1.7.6

\left(\int_{a}^{b}(f(x)+g(x))^{p}\,\mathrm{d}x\right)^{1/p}\leq\left(\int_{a}^% {b}(f(x))^{p}\,\mathrm{d}x\right)^{1/p}+\left(\int_{a}^{b}(g(x))^{p}\,\mathrm{% d}x\right)^{1/p}.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $p>1$
A&S Ref:: 3.2.13
Permalink:: http://dlmf.nist.gov/1.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(ii), §1.7(ii), §1.7 and Ch.1

… ►

1.7.8

\min(a_{1},a_{2},\dots,a_{n})\leq M(r)\leq\max(a_{1},a_{2},\dots,a_{n}),

ⓘ

Symbols:: $n$ : nonnegative integer and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.2.2
Permalink:: http://dlmf.nist.gov/1.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(iii), §1.7 and Ch.1

…

… ►In

\mathbb{C}

both the modulus and phase of the asymptotic variable

z

need to be taken into account. … ►In the transition through

\theta=\pi

,

\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)

changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case

\rho=100

. … ►Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines (

\theta=\pi

in the present example). … ►(This means that, if necessary,

z

is replaced by

z/(\lambda_{2}-\lambda_{1})

.) … ►The process just used is equivalent to re-expanding the remainder term of the original asymptotic series (2.11.24) in powers of

1/(x+5)

and truncating the new series optimally. …

… ►

J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

ⓘ

External Links:: Document, MathReview (R. D. James), ZentralBlatt
Referenced by:: §17.18.
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.

ⓘ

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

… ►

D. A. Levine (1969) Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 8S. For Remarks see same journal 13(1970), pp. 124 and 449.
External Links:: ISSN 0001-0782, Document, Remark 1 and Remark 2
Referenced by:: §8.28(iv).
Encodings:: BibTeX

… ►

H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iv).
Encodings:: BibTeX

…

… ►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument

z

and the modulus

k

is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►Given real or complex numbers

a_{0},b_{0}

, with

b_{0}/a_{0}

not real and negative, define … ►From (22.7.1),

k_{1}=\tfrac{1}{19}

and

x/(1+k_{1})=0.19

. … ►If either

\tau

or

q=e^{i\pi\tau}

is given, then we use

k={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)

,

k^{\prime}={\theta_{4}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)

,

K=\frac{1}{2}\pi{\theta_{3}}^{2}\left(0,q\right)

, and

K^{\prime}=-i\tau K

, obtaining the values of the theta functions as in §20.14. … ►If

k=k^{\prime}=1/\sqrt{2}

, then three iterations of (22.20.1) give

M=0.84721\;30848

, and from (22.20.6)

K=\pi/(2M)=1.85407\;46773

— in agreement with the value of

\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{2}/\left(4\sqrt{\pi}\right)

; compare (23.17.3) and (23.22.2). …

… ►

T. Weider (1999) Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt
Referenced by:: 7th item, 10th item.
Encodings:: BibTeX

… ►

J. Wimp (1985) Some explicit Padé approximants for the function

\Phi^{\prime}/\Phi

and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt
Referenced by:: §13.31(ii).
Encodings:: BibTeX

… ►

M. E. Wojcicki (1961) Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

ⓘ

Notes:: Includes Algol code, maximum accuracy 8S.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(ii).
Encodings:: BibTeX

…

… ►

18.14.3

\left(\tfrac{1}{2}(1-x)\right)^{\frac{1}{2}\alpha+\frac{1}{4}}\left(\tfrac{1}{% 2}(1+x)\right)^{\frac{1}{2}\beta+\frac{1}{4}}|P^{(\alpha,\beta)}_{n}\left(x% \right)|\leq\frac{\Gamma\left(\max(\alpha,\beta)+n+1\right)}{{\pi}^{\frac{1}{2% }}n!\left(n+\tfrac{1}{2}(\alpha+\beta+1)\right)^{\max(\alpha,\beta)+\frac{1}{2% }}},

-1\leq x\leq 1

,

-\tfrac{1}{2}\leq\alpha\leq\tfrac{1}{2}

,

-\tfrac{1}{2}\leq\beta\leq\tfrac{1}{2}

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(i), §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

►

18.14.3_5

\left(\tfrac{1}{2}(1+x)\right)^{\beta/2}\left|P^{(\alpha,\beta)}_{n}\left(x% \right)\right|\leq P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left(\alpha+1% \right)_{n}}}{n!},

-1\leq x\leq 1

,

\alpha,\beta\geq 0

.

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Proved:: Carnicer et al. (2020, Theorem 2)(proved)
Proof sketch:: The case $\alpha=0$ follows from the result on monotonic weight functions in §18.2(xii).
Referenced by:: §18.14(i), §18.14(i), §18.2(xii), Erratum (V1.2.0) Section 18.14
Permalink:: http://dlmf.nist.gov/18.14.E3_5
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

… ►Let

R_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)/P^{(\alpha,\beta)}_{n}\left(1\right)

. … ►

18.14.15

x_{n,m}\leq(\beta-\alpha)/(\alpha+\beta+1)\leq x_{n,m+1}.

ⓘ

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.14.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

…

… ►

19.34.1

\frac{{c}^{2}M}{2\pi}=ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos\theta)^{-1/2% }\cos\theta\,\mathrm{d}\theta=2ab\int_{-1}^{1}\frac{t\,\mathrm{d}t}{\sqrt{(1+t% )(1-t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5}),

ⓘ

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, $I(\mathbf{m})$ : integral, $M$ : mutual inductance and $h$ : distance
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.34.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.34 and Ch.19

… ►is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. … ►

19.34.7

M=(2/{c}^{2})(\pi a^{2})(\pi b^{2})R_{-\frac{3}{2}}\left(\tfrac{3}{2},\tfrac{3% }{2};r_{+}^{2},r_{-}^{2}\right).

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $M$ : mutual inductance and $r_{\pm}$
Referenced by:: §19.34
Permalink:: http://dlmf.nist.gov/19.34.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.34 and Ch.19

…

… ►

J. Segura and A. Gil (1998) Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

ⓘ

Notes:: Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

W. V. Snyder (1993) Algorithm 723: Fresnel integrals. ACM Trans. Math. Software 19 (4), pp. 452–456.

ⓘ

Notes:: Single- and double-precision Fortran, maximum accuracy 16D. For remarks see same journal v. 22 (1996), p. 498-500 and v. 26 (2000), p. 617
External Links:: ISSN 0098-3500, Document, GAMS (module 723; package TOMS), ZentralBlatt
Referenced by:: 3rd item, 1st item.
Encodings:: BibTeX

… ►

B. Sommer and J. G. Zabolitzky (1979) On numerical Bessel transformation. Comput. Phys. Comm. 16 (3), pp. 383–387.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 15D.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 8th item.
Encodings:: BibTeX

… ►

I. A. Stegun and R. Zucker (1970) Automatic computing methods for special functions. I. J. Res. Nat. Bur. Standards Sect. B 74B, pp. 211–224.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 18S.
External Links:: MathReview (Bernard H. Rosman), ZentralBlatt
Referenced by:: §7.25(ii).
Encodings:: BibTeX

… ►

A. J. Stone and C. P. Wood (1980) Root-rational-fraction package for exact calculation of vector-coupling coefficients. Comput. Phys. Comm. 21 (2), pp. 195–205.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 16D
External Links:: Document, Code
Referenced by:: 10th item.
Encodings:: BibTeX

…

… ►

31.8.3

g=\tfrac{1}{2}\max\left(2\max_{0\leq k\leq 3}m_{k},1+N-(1+(-1)^{N})\left(% \tfrac{1}{2}+\min_{0\leq k\leq 3}m_{k}\right)\right).

ⓘ

Symbols:: $m$ : nonnegative integer, $g$ : degree of $\lambda$ and $N$ : degree of $z$
Permalink:: http://dlmf.nist.gov/31.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.8 and Ch.31

… ►

\Psi_{1,-1}=\left(z^{2}+(\lambda+3a+3)z+a\right)/z^{3},

…

maximum modulus

21—30 of 47 matching pages

21: 2.5 Mellin Transform Methods

22: 1.7 Inequalities

23: 2.11 Remainder Terms; Stokes Phenomenon

24: Bibliography L

25: 22.20 Methods of Computation

26: Bibliography W

27: 18.14 Inequalities

28: 19.34 Mutual Inductance of Coaxial Circles

29: Bibliography S

30: 31.8 Solutions via Quadratures