# ������2���������������WeChat���aptao168���Vk8YcIz

(0.002 seconds)

## 1—10 of 798 matching pages

##### 1: 4.17 Special Values and Limits
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 2: 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. …
34.5.5 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}\end{Bmatrix}=(-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-% 2j_{3}+1)}{2j_{2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac% {1}{2}},$
34.5.6 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}+1\end{Bmatrix}=(-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j% _{3}+1)(J-2j_{3}+2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}% \right)^{\frac{1}{2}},$
34.5.7 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}&j_{2}\end{Bmatrix}=(-1)^{J+1}\frac{2(j_{2}(j_{2}+1)+j_{3}(j_{3}+1)-j_{% 1}(j_{1}+1))}{\left(2j_{2}(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)(2j_{3}+2)\right% )^{\frac{1}{2}}}.$
34.5.13 $E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.$
##### 3: 29.1 Special Notation
The main functions treated in this chapter are the eigenvalues $a^{2m}_{\nu}\left(k^{2}\right)$, $a^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+2}_{\nu}\left(k^{2}\right)$, the Lamé functions $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$, and the Lamé polynomials $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$, $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$. … Other notations that have been used are as follows: Ince (1940a) interchanges $a^{2m+1}_{\nu}\left(k^{2}\right)$ with $b^{2m+1}_{\nu}\left(k^{2}\right)$. The relation to the Lamé functions $L^{(m)}_{c\nu}$, $L^{(m)}_{s\nu}$of Jansen (1977) is given by …
$\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=s_{\nu}^{2m+2}(k^{2}){\rm Es}_{% \nu}^{2m+2}(z,k^{2}),$
where the positive factors $c_{\nu}^{m}(k^{2})$ and $s_{\nu}^{m}(k^{2})$ are determined by …
##### 6: 22.9 Cyclic Identities
###### §22.9(ii) Typical Identities of Rank 2
These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}s_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1$; $n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1$. …
22.9.11 $\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{2}d_% {1,2}^{(2)}=k^{\prime}\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$
22.9.12 $c_{1,2}^{(2)}s_{1,2}^{(2)}d_{2,2}^{(2)}+c_{2,2}^{(2)}s_{2,2}^{(2)}d_{1,2}^{(2)% }=0.$
22.9.21 $k^{2}c_{1,2}^{(2)}s_{1,2}^{(2)}c_{2,2}^{(2)}s_{2,2}^{(2)}=k^{\prime}\left(1-% \left(s_{1,2}^{(2)}\right)^{2}-\left(s_{2,2}^{(2)}\right)^{2}\right).$
##### 7: 30.3 Eigenvalues
With $\mu=m=0,1,2,\dots$, the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form $(1-x^{2})^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. These solutions exist only for eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$, $n=m,m+1,m+2,\dots$, of the parameter $\lambda$. … The eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ are analytic functions of the real variable $\gamma^{2}$ and satisfy … has the solutions $\lambda=\lambda^{m}_{m+2j}\left(\gamma^{2}\right)$, $j=0,1,2,\dots$. If $p$ is an odd positive integer, then Equation (30.3.5) has the solutions $\lambda=\lambda^{m}_{m+2j+1}\left(\gamma^{2}\right)$, $j=0,1,2,\dots$. …
##### 8: 14.29 Generalizations
14.29.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{% \mu_{2}^{2}}{2(1+z)}\right)w}=0$
As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when $\mu_{1}=\mu_{2}=\mu$. …
##### 9: 29.6 Fourier Series
###### §29.6(i) Function $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$
In the special case $\nu=2n$, $m=0,1,\dots,n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\dots$. …
##### 10: 29.12 Definitions
where $n=0,1,2,\dots$, $m=0,1,2,\dots,n$. … In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\operatorname{sn}}^{2})$ denotes a polynomial of degree $n$ in ${\operatorname{sn}}^{2}\left(z,k\right)$ (different for each type). … where $\rho$, $\sigma$, $\tau$ are either $0$ or $\frac{1}{2}$. The polynomial $P(\xi)$ is of degree $n$ and has $m$ zeros (all simple) in $(0,1)$ and $n-m$ zeros (all simple) in $(1,k^{-2})$. … defined for $(t_{1},t_{2},\dots,t_{n})$ with …