# half argument

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## 11—20 of 22 matching pages

##### 11: 34.2 Definition: $\mathit{3j}$ Symbol
Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. …
34.2.2 $m_{r}=-j_{r},-j_{r}+1,\dots,j_{r}-1,j_{r},$ $r=1,2,3$,
and the summation is over all nonnegative integers $s$ such that the arguments in the factorials are nonnegative. … For alternative expressions for the $\mathit{3j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{3}F_{2}}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
##### 12: 34.6 Definition: $\mathit{9j}$ Symbol
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
34.6.2 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$
The $\mathit{9j}$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
##### 13: 34.7 Basic Properties: $\mathit{9j}$ Symbol
34.7.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.$
Odd permutations of columns or rows introduce a phase factor $(-1)^{R}$, where $R$ is the sum of all arguments of the $\mathit{9j}$ symbol. …
34.7.2 $\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.$
34.7.3 $\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\end{Bmatrix}.$
34.7.5 $\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\end{Bmatrix}.$
##### 14: 34.4 Definition: $\mathit{6j}$ Symbol
34.4.1 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m_{r}m^{\prime}_{s}}(-1)^{l_{1}+m^{\prime% }_{1}+l_{2}+m^{\prime}_{2}+l_{3}+m^{\prime}_{3}}\*\begin{pmatrix}j_{1}&j_{2}&j% _{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix},$
34.4.2 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},$
where the summation is over all nonnegative integers $s$ such that the arguments in the factorials are nonnegative. …
34.4.3 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),$
For alternative expressions for the $\mathit{6j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{4}F_{3}}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
##### 15: 19.23 Integral Representations
19.23.6 $4\pi R_{F}\left(x,y,z\right)=\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\frac{\sin% \theta\mathrm{d}\theta\mathrm{d}\phi}{(x{\sin}^{2}\theta{\cos}^{2}\phi+y{\sin}% ^{2}\theta{\sin}^{2}\phi+z{\cos}^{2}\theta)^{1/2}},$
19.23.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4\Gamma\left(b_{1}+b_{2}+b_{3}% \right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\Gamma\left(b_{3}% \right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left(\sum_{j=1}^{3}z_{j}l_{j}% ^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin\theta\mathrm{d}\theta% \mathrm{d}\phi,$ $b_{j}>0$, $\Re z_{j}>0$.
19.23.10 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{1}u^{a-1}(1-u)^{a^{\prime}-1}\*\prod_{j=1}^{n}(1-u+uz_{j})^{% -b_{j}}\mathrm{d}u,$ $a,a^{\prime}>0$; $a+a^{\prime}=\sum_{j=1}^{n}b_{j}$; $z_{j}\in\mathbb{C}\setminus(-\infty,0]$.
##### 16: 15.19 Methods of Computation
For $z\in\mathbb{R}$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. For $z\in\mathbb{C}$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm\pi\mathrm{i}/3}$. This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi\mathrm{i}/3},{\mathrm{e}}^{-\pi\mathrm{i}/3}\}$ onto itself. … For example, in the half-plane $\Re z\leq\frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F\left(a,b;c+N+1;z\right)$ and $F\left(a,b;c+N;z\right)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …
##### 17: 19.16 Definitions
19.16.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n}_{j=1}(t+z_{j})^{-b_{j}}% \mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a% -1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\mathrm{d}t,$ $b_{1}+\cdots+b_{n}>a>0$, $b_{j}\in\mathbb{R}$, $z_{j}\in\mathbb{C}\setminus(-\infty,0]$,
19.16.12 $R_{-a}\left(b_{1},\dots,b_{4};c-1,c-k^{2},c,c-\alpha^{2}\right)=\frac{2({\sin}% ^{2}\phi)^{1-a^{\prime}}}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\phi}(% \sin\theta)^{2a-1}{({\sin}^{2}\phi-{\sin}^{2}\theta)}^{a^{\prime}-1}\*(\cos% \theta)^{1-2b_{1}}{(1-k^{2}{\sin}^{2}\theta)}^{-b_{2}}{(1-\alpha^{2}{\sin}^{2}% \theta)}^{-b_{4}}\mathrm{d}\theta,$
$R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime},b_{1},\dots,b_{n}$ are half-odd-integers, the rest are integers, and none of $a$, $a^{\prime}$, $a+a^{\prime}$ is zero or a negative integer. …
##### 18: Bibliography G
• W. Gautschi (1965) Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.
• W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
• A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.
• A. Gil and J. Segura (1997) Evaluation of Legendre functions of argument greater than one. Comput. Phys. Comm. 105 (2-3), pp. 273–283.
• A. Gil, J. Segura, and N. M. Temme (2004a) Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.
• ##### 19: Bibliography P
• B. V. Pal${}^{\prime}$tsev (1999) On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).
• R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
• S. Paszkowski (1991) Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.
• ##### 20: 4.37 Inverse Hyperbolic Functions
Graphs of the principal values for real arguments are given in §4.29. This section also indicates conformal mappings, and surface plots for complex arguments. …
4.37.17 $\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^{% 1/2}+y\right),$ $y\in[1,\infty)$,
4.37.22 $\operatorname{arccosh}x=\pm\ln\left(i(1-x^{2})^{1/2}+x\right),$ $x\in(-1,1]$,
4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;