DLMF: Untitled Document

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British Association for the Advancement of Science (1937) tabulates $I_{0}\left(x\right)$ , $I_{1}\left(x\right)$ , $x=0(.001)5$ , 7–8D; $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ , $x=0.01(.01)5$ , 7–10D; $e^{-x}I_{0}\left(x\right)$ , $e^{-x}I_{1}\left(x\right)$ , $e^{x}K_{0}\left(x\right)$ , $e^{x}K_{1}\left(x\right)$ , $x=5(.01)10(.1)20$ , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ for small values of $x$ .

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Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

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The main tables in Abramowitz and Stegun (1964, Chapter 9) give $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)10(.2)20$ , 8D–10D or 10S; $\sqrt{x}e^{-x}I_{n}\left(x\right)$ , $(\sqrt{x}/\pi)$ $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $1/x=0(.002)0.05$ ; $K_{0}\left(x\right)+I_{0}\left(x\right)\ln x$ , $x(K_{1}\left(x\right)-I_{1}\left(x\right)\ln x)$ , $x=0(.1)2$ , 8D; $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)10(.5)20$ , 5S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 9–10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

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Zhang and Jin (1996, p. 271) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)1(.5)20$ , 8D.

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… 笆ｺThompson (1997, pp. 499, 504) uses descending Landen transformations for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

. …For computation of

K\left(k\right)

and

E\left(k\right)

with complex

k

see Fettis and Caslin (1969) and Morita (1978). … 笆ｺLee (1990) compares the use of theta functions for computation of

K\left(k\right)

,

E\left(k\right)

, and

K\left(k\right)-E\left(k\right)

,

0\leq k^{2}\leq 1

, with four other methods. …For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). For integrals of the second and third kinds see Lawden (1989, §§3.4–3.7). …

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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	Open Source									With Book				Commercial
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14.34(ii) $\mathsf{P}_{\nu}\left(x\right)$ , $\mathsf{Q}_{\nu}\left(x\right)$ , $P_{\nu}\left(x\right)$ , $Q_{\nu}\left(x\right)$ , $x,\nu\in\mathbb{R}$	笨�		笨�	笨�			笨�	a	笨�	笨�	笨�	笨�	笨�		笨�	笨�	笨�	笨�
14.34(iii) $\mathsf{P}_{\nu}\left(z\right)$ , $\mathsf{Q}_{\nu}\left(z\right)$ , $P_{\nu}\left(z\right)$ , $Q_{\nu}\left(z\right)$ , $z,\nu\in\mathbb{C}$	笨�							a	笨�	笨�					笨�	笨�	a
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19.39(ii) $K\left(k\right)$ , $E\left(k\right)$ , $0\leq k^{2}\leq 1$	笨�	笨�	笨�	笨�	笨�	笨�		笨�	笨�					笨�	笨�	笨�	笨�
19.39(iii) $F\left(\phi,k\right)$ , $E\left(\phi,k\right)$ , $\phi,k\in\mathbb{R}$	笨�	笨�	笨�	笨�	笨�	笨�		笨�	笨�		笨�				笨�	笨�	a	笨�
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20 Theta Functions
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… 笆ｺand two similar formulas by symmetry; compare the second row in Table 18.6.1. … 笆ｺ

T_{5}\left(x\right)=16x^{5}-20x^{3}+5x,

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U_{0}\left(x\right)=1,

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U_{1}\left(x\right)=2x,

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U_{2}\left(x\right)=4x^{2}-1,

…

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Y_{l}^{m}\left(\theta,\phi\right)

are known as surface harmonics of the first kind: tesseral for

|m|<l

and sectorial for

|m|=l

. … 笆ｺ

P^{m}_{n}\left(x\right)

and

Q^{m}_{n}\left(x\right)

(

x>1

) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively.

P^{m}_{n}\left(ix\right)

and

Q^{m}_{n}\left(ix\right)

(

x>0

) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics

R_{n}^{m}(x)=e^{-i\pi n/2}P^{m}_{n}\left(ix\right)

and

T_{n}^{m}(x)=ie^{i\pi n/2}Q^{m}_{n}\left(ix\right)

which are real when

x>0

and

n=0,1,2,\dots

. … 笆ｺMost mathematical properties of

Y_{{l},{m}}\left(\theta,\phi\right)

can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. …

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A. Laforgia (1979) On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §10.21(vi).
Encodings:: BibTeX

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P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

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D. K. Lee (1990) Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds. Comput. Phys. Comm. 60 (3), pp. 319–327.

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External Links:: ISSN 0010-4655, Document, MathReview, ZentralBlatt
Referenced by:: §19.36(iii), §19.5.
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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N. A. Lukaﾅ｡eviﾄ� (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:: In Russian. English translation: Differential Equations 7(6), pp. 853–854
External Links:: MathReview (Z. Rubinstein), ZentralBlatt
Referenced by:: §32.7(ii).
Encodings:: BibTeX

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… 笆ｺConsequently, Bessel functions

J_{n}\left(x\right)

, and modified Bessel functions

I_{n}\left(x\right)

, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … 笆ｺOn separation of variables into cylindrical coordinates, the Bessel functions

J_{n}\left(x\right)

, and modified Bessel functions

I_{n}\left(x\right)

and

K_{n}\left(x\right)

, all appear. … 笆ｺThe functions

\mathsf{j}_{n}\left(x\right)

,

\mathsf{y}_{n}\left(x\right)

,

{\mathsf{h}^{(1)}_{n}}\left(x\right)

, and

{\mathsf{h}^{(2)}_{n}}\left(x\right)

arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates

\rho,\theta,\phi

(§1.5(ii)): …With the spherical harmonic

Y_{\ell,m}\left(\theta,\phi\right)

defined as in §14.30(i), the solutions are of the form

f=g_{\ell}(k\rho)Y_{\ell,m}\left(\theta,\phi\right)

with

g_{\ell}=\mathsf{j}_{\ell}

,

\mathsf{y}_{\ell}

,

{\mathsf{h}^{(1)}_{\ell}}

, or

{\mathsf{h}^{(2)}_{\ell}}

, depending on the boundary conditions. …

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D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

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E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

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E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

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of%20the%20second%20kind

1—10 of 20 matching pages

1: 10.75 Tables

2: 19.36 Methods of Computation

3: Bibliography M

4: Bibliography F

5: Software Index

6: 18.5 Explicit Representations

7: 14.30 Spherical and Spheroidal Harmonics

8: Bibliography L

9: 10.73 Physical Applications

10: Bibliography N