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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Zhang and Jin (1996, pp. 240–250) tabulates $I_{n}\left(x\right)$ , $I_{n}'\left(x\right)$ , $K_{n}\left(x\right)$ , $K_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1,5,10,25,50,100$ , 9S; $I_{n+\alpha}\left(x\right)$ , $I_{n+\alpha}'\left(x\right)$ , $K_{n+\alpha}\left(x\right)$ , $K_{n+\alpha}'\left(x\right)$ , $n=0(1)5$ , 10, 30, 50, 100, $\alpha=\tfrac{1}{4}$ , $\tfrac{1}{3}$ , $\tfrac{1}{2}$ , $\tfrac{2}{3}$ , $\tfrac{3}{4}$ , $x=1$ , 5, 10, 50, 8S; real and imaginary parts of $I_{n+\alpha}\left(z\right)$ , $I_{n+\alpha}'\left(z\right)$ , $K_{n+\alpha}\left(z\right)$ , $K_{n+\alpha}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $\alpha=0,\tfrac{1}{2}$ , $z=4+2i,20+10i$ , 8S.

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Kerimov and Skorokhodov (1984c) tabulates all zeros of $I_{-n-\frac{1}{2}}\left(z\right)$ and $I_{-n-\frac{1}{2}}'\left(z\right)$ in the sector $0\leq\operatorname{ph}z\leq\tfrac{1}{2}\pi$ for $n=1(1)20$ , 9S.

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

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36.2.28

\Psi^{(\mathrm{E})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{E})}\left(0,0,-% z\right)}\\ =2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6% }\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right),

z\geq 0

,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit and $z$ : real parameter
Referenced by:: Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/36.2.E28
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. For the proof see Berry and Howls (2010).
See also:: Annotations for §36.2(iv), §36.2 and Ch.36

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See accompanying text — Figure 10.3.10: ${H^{(1)}_{0}}\left(x+iy\right)$ , $-10\leq x\leq 5$ , $-2.8\leq y\leq 4$ . … Magnify 3D Help

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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

. … ►

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

. … ►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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F\left(\phi,k\right)

can be evaluated by using (19.25.5). …A summary for

F\left(\phi,k\right)

is given in Gautschi (1975, §3). … ►Also, see Todd (1975) for a special case of

K\left(k\right)

. 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 fast methods for computing the incomplete elliptic integral of the first kind see Karp and Sitnik (2007) and Fukushima (2010). …

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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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… ►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … ►Bessel functions of the first kind,

J_{n}\left(x\right)

, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation

\nabla^{2}V=0

, or by the Helmholtz equation

(\nabla^{2}+k^{2})\psi=0

. … ►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). … ►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. …

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Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Zhang and Jin (1996) tabulates $\mathbf{H}_{n}\left(x\right)$ and $\mathbf{L}_{n}\left(x\right)$ for $n=-4(1)3$ and $x=0(1)20$ to 8D or 7S.

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Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

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	Open Source									With Book				Commercial
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14.34(ii) $\mathsf{P}_{\nu}\left(x\right)$ , $\mathsf{Q}_{\nu}\left(x\right)$ , $P_{\nu}\left(x\right)$ , $Q_{\nu}\left(x\right)$ , $x,\nu\in\mathbb{R}$	✓		✓	✓			✓	a	✓	✓	✓	✓	✓		✓	✓	✓	✓
14.34(iii) $\mathsf{P}_{\nu}\left(z\right)$ , $\mathsf{Q}_{\nu}\left(z\right)$ , $P_{\nu}\left(z\right)$ , $Q_{\nu}\left(z\right)$ , $z,\nu\in\mathbb{C}$	✓							a	✓	✓					✓	✓	a
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19.39(ii) $K\left(k\right)$ , $E\left(k\right)$ , $0\leq k^{2}\leq 1$	✓	✓	✓	✓	✓	✓		✓	✓					✓	✓	✓	✓
19.39(iii) $F\left(\phi,k\right)$ , $E\left(\phi,k\right)$ , $\phi,k\in\mathbb{R}$	✓	✓	✓	✓	✓	✓		✓	✓		✓				✓	✓	a	✓
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20 Theta Functions
…

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of%20the%20first%20kind

1—10 of 26 matching pages

1: 10.75 Tables

2: 18.5 Explicit Representations

3: 36.2 Catastrophes and Canonical Integrals

4: 10.3 Graphics

5: 14.30 Spherical and Spheroidal Harmonics

6: 19.36 Methods of Computation

7: Bibliography F

8: 10.73 Physical Applications

9: 11.14 Tables

10: Software Index