half argument

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1—10 of 22 matching pages

2: Bibliography F

… ►

T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

ⓘ

External Links:: ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt
Referenced by:: §19.36(iv), §19.36(iv).
Encodings:: BibTeX

…

4: 20.2 Definitions and Periodic Properties

… ►

§20.2(iii) Translation of the Argument by Half-Periods

…

	Open Source	With Book	Commercial
…
10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM, NMS
…
10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions	✓	✓	✓								a	✓	✓			✓	✓	✓	✓	✓	✓	✓
…

6: Bibliography S

… ►

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

…

L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1988a) Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

ⓘ

Notes:: English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117
External Links:: ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.77(iv).
Encodings:: BibTeX

…

8: 14.34 Software

… ►

§14.34(ii) Legendre Functions: Real Argument and Parameters

… ►

•

Gil and Segura (1997). Integer and half-integer parameters. Fortran.

… ►

§14.34(iii) Legendre Functions: Complex Argument and/or Parameters

►

•

Gil and Segura (1998). Integer parameters and purely imaginary arguments. Fortran.

…

9: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The Coulomb functions given in this chapter are most commonly evaluated for real values of

\rho

r

\eta

\epsilon

and nonnegative integer values of

\ell

, but they may be continued analytically to complex arguments and order

\ell

as indicated in §33.13. … ►

•

Searches for resonances as poles of the $S$ -matrix in the complex half-plane $\Im\sf{k}<0$ . See for example Csótó and Hale (1997).

…

10: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►If any lower argument in a

\mathit{6j}

symbol is

0

\tfrac{1}{2}

, or

1

, then the

\mathit{6j}

symbol has a simple algebraic form. … ►The

\mathit{6j}

symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example, … ►

34.5.13

E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.

ⓘ

Defines:: $E(j)$ (locally)
Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

… ►

34.5.19

\sum_{l}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}=0,

2\mu-j

odd,

\mu=\min(j_{1},j_{2})

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

►

34.5.20

\sum_{l}(-1)^{l+j}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{1}&j_{2}&j\end{Bmatrix}=\frac{(-1)^{2\mu}}{2j+1},

\mu=\min(j_{1},j_{2})

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

…

half argument

1—10 of 22 matching pages

1: 22.6 Elementary Identities

§22.6(iii) Half Argument

2: Bibliography F

3: 10.77 Software

§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions

4: 20.2 Definitions and Periodic Properties

§20.2(iii) Translation of the Argument by Half-Periods

5: Software Index

6: Bibliography S

7: Bibliography B

8: 14.34 Software

§14.34(ii) Legendre Functions: Real Argument and Parameters

§14.34(iii) Legendre Functions: Complex Argument and/or Parameters

9: 33.22 Particle Scattering and Atomic and Molecular Spectra

10: 34.5 Basic Properties: $\mathit{6j}$ Symbol

half argument

1—10 of 22 matching pages

1: 22.6 Elementary Identities

§22.6(iii) Half Argument

2: Bibliography F

3: 10.77 Software

§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions

4: 20.2 Definitions and Periodic Properties

§20.2(iii) Translation of the Argument by Half-Periods

5: Software Index

6: Bibliography S

7: Bibliography B

8: 14.34 Software

§14.34(ii) Legendre Functions: Real Argument and Parameters

§14.34(iii) Legendre Functions: Complex Argument and/or Parameters

9: 33.22 Particle Scattering and Atomic and Molecular Spectra

10: 34.5 Basic Properties: 6 ⁢ j Symbol

10: 34.5 Basic Properties: $\mathit{6j}$ Symbol