spherical trigonometry

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11: 10.60 Sums

§10.60 Sums

►

§10.60(i) Addition Theorems

… ►

§10.60(ii) Duplication Formulas

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►

§10.60(iv) Compendia

…

12: Bibliography H

… ►

F. E. Harris (2000) Spherical Bessel expansions of sine, cosine, and exponential integrals. Appl. Numer. Math. 34 (1), pp. 95–98.

ⓘ

External Links:: ISSN 0168-9274, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii), §6.10(ii), §6.15.
Encodings:: BibTeX

… ►

E. W. Hobson (1928) A Treatise on Plane and Advanced Trigonometry. 7th edition, Cambridge University Press.

ⓘ

Notes:: Reprinted by Dover Publications Inc., New York, 2004.
External Links:: MathReview, ZentralBlatt
Referenced by:: §4.16, §4.17, §4.19, §4.2(ii), §4.2(iii), §4.21(i), §4.21(ii), §4.21(iii), §4.21(iv), §4.22, §4.23(v), §4.23(vi), §4.24(i), §4.24(iii), §4.28, §4.30, §4.31, §4.35(i), §4.35(ii), §4.35(iii), §4.35(iv), §4.42(i), §4.42(ii), §4.43, §4.8(ii), Ch.4.
Encodings:: BibTeX

►

E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.

ⓘ

Notes:: Reprinted by Chelsea Publishing Company, New York, 1955 and 1965.
External Links:: ISBN 0-8284-0104-7, MathReview, ZentralBlatt
Referenced by:: §14.1, §14.11, §14.16(ii), §14.16(iii), §14.20(x), §14.27, §14.27, §14.3(i), §14.3(i), Ch.14, §29.18(iii), Ch.29.
Encodings:: BibTeX

…

13: 10.51 Recurrence Relations and Derivatives

… ►Let

f_{n}(z)

denote any of

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

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

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

, or

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

. … ►

nf_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_{n}^{\prime}(z),

n=1,2,\dots

… ►

f_{n}^{\prime}(z)=-f_{n+1}(z)+(n/z)f_{n}(z),

n=0,1,\dots.

… ►Let

g_{n}(z)

denote

{\mathsf{i}^{(1)}_{n}}\left(z\right)

{\mathsf{i}^{(2)}_{n}}\left(z\right)

, or

(-1)^{n}

\mathsf{k}_{n}\left(z\right)

. Then …

14: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for

\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)

\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)

{\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)

{\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)

{\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)

, and

\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)

n\to\infty

that are uniform with respect to

z

can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for

{\mathsf{i}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)

the connection formula (10.47.11) is available. ►For the corresponding expansion for

\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)

use ►

10.57.1

\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)=\frac{\pi^{\frac{1}{2}}}{((2n+1)% z)^{\frac{1}{2}}}J_{n+\frac{1}{2}}'\left((n+\tfrac{1}{2})z\right)-\frac{\pi^{% \frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}J_{n+\frac{1}{2}}\left((n+\tfrac{1}{2})z% \right).

ⓘ

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, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
Referenced by:: §10.57
Permalink:: http://dlmf.nist.gov/10.57.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.57 and Ch.10

…

15: 10.58 Zeros

§10.58 Zeros

►For

n\geq 0

the

m

th positive zeros of

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

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

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

, and

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

are denoted by

a_{n,m}

a^{\prime}_{n,m}

b_{n,m}

, and

b^{\prime}_{n,m}

, respectively, except that for

n=0

we count

x=0

as the first zero of

\mathsf{j}_{0}'\left(x\right)

. … ►

\mathsf{j}_{n}'\left(a_{n,m}\right)=\sqrt{\frac{\pi}{2j_{n+\frac{1}{2},m}}}J_{% n+\frac{1}{2}}'\left(j_{n+\frac{1}{2},m}\right),

►

\mathsf{y}_{n}'\left(b_{n,m}\right)=\sqrt{\frac{\pi}{2y_{n+\frac{1}{2},m}}}Y_{% n+\frac{1}{2}}'\left(y_{n+\frac{1}{2},m}\right).

…

16: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions

J_{\nu}\left(z\right)

Y_{\nu}\left(z\right)

; Hankel functions

{H^{(1)}_{\nu}}\left(z\right)

{H^{(2)}_{\nu}}\left(z\right)

; modified Bessel functions

I_{\nu}\left(z\right)

K_{\nu}\left(z\right)

; spherical Bessel functions

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

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

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

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

; modified spherical Bessel functions

{\mathsf{i}^{(1)}_{n}}\left(z\right)

{\mathsf{i}^{(2)}_{n}}\left(z\right)

\mathsf{k}_{n}\left(z\right)

; Kelvin functions

\operatorname{ber}_{\nu}\left(x\right)

\operatorname{bei}_{\nu}\left(x\right)

\operatorname{ker}_{\nu}\left(x\right)

\operatorname{kei}_{\nu}\left(x\right)

. For the spherical Bessel functions and modified spherical Bessel functions the order

n

is a nonnegative integer. … ►Abramowitz and Stegun (1964):

j_{n}(z)

y_{n}(z)

h_{n}^{(1)}(z)

h_{n}^{(2)}(z)

, for

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

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

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

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

, respectively, when

n\geq 0

. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

17: 10.54 Integral Representations

§10.54 Integral Representations

… ►

10.54.2

\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P% _{n}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta.

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.14
Permalink:: http://dlmf.nist.gov/10.54.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

►

10.54.3

\mathsf{k}_{n}\left(z\right)=\frac{\pi}{2}\int_{1}^{\infty}e^{-zt}P_{n}\left(t% \right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi.

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.54.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

… ►

{\mathsf{h}^{(1)}_{n}}\left(z\right)=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+% )}e^{izt}Q_{n}\left(t\right)\,\mathrm{d}t,

… ►For the Legendre polynomial

P_{n}

and the associated Legendre function

Q_{n}

see §§18.3 and 14.21(i), with

\mu=0

and

\nu=n

. …

18: 6.10 Other Series Expansions

… ►

§6.10(ii) Expansions in Series of Spherical Bessel Functions

… ►

6.10.4

\operatorname{Si}\left(z\right)=z\sum_{n=0}^{\infty}\left(\mathsf{j}_{n}\left(% \tfrac{1}{2}z\right)\right)^{2},

ⓘ

Symbols:: $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §6.10(ii)
Permalink:: http://dlmf.nist.gov/6.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §6.10(ii), §6.10 and Ch.6

►

6.10.5

\operatorname{Cin}\left(z\right)=\sum_{n=1}^{\infty}a_{n}\left(\mathsf{j}_{n}% \left(\tfrac{1}{2}z\right)\right)^{2},

ⓘ

Symbols:: $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $z$ : complex variable, $n$ : nonnegative integer and $a_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/6.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §6.10(ii), §6.10 and Ch.6

►

6.10.6

\operatorname{Ei}\left(x\right)=\gamma+\ln\left|x\right|+\sum_{n=0}^{\infty}(-% 1)^{n}(x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}x\right)\right)^{% 2},

x\neq 0

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\ln\NVar{z}$ : principal branch of logarithm function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $x$ : real variable, $n$ : nonnegative integer and $a_{n}$ : coefficients
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §6.10(ii), §6.10 and Ch.6

… ►

6.10.8

\operatorname{Ein}\left(z\right)=ze^{-z/2}\left({\mathsf{i}^{(1)}_{0}}\left(% \tfrac{1}{2}z\right)+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1)}{\mathsf{i}^{(1)}_% {n}}\left(\tfrac{1}{2}z\right)\right).

ⓘ

Symbols:: $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §6.10(ii)
Permalink:: http://dlmf.nist.gov/6.10.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §6.10(ii), §6.10 and Ch.6

…

19: 10.73 Physical Applications

… ► … ►

§10.73(ii) Spherical Bessel Functions

►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. Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …

20: 10.59 Integrals

§10.59 Integrals

►

10.59.1

\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t=\begin% {cases}\pi i^{n}P_{n}\left(b\right),&-1<b<1,\\ \frac{1}{2}\pi(\pm\mathrm{i})^{n},&b=\pm 1,\\ 0,&\pm b>1,\end{cases}

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 11.4.26
Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.59.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.59 and Ch.10

►where

P_{n}

is the Legendre polynomial (§18.3). ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …