# 鉄鹿山学院大学学士成绩单【仿证微CXFK69】2Efze

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##### 1: 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 …
##### 3: 4.17 Special Values and Limits
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 4: 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 $k^{2}c_{1,2}^{(2)}s_{1,2}^{(2)}c_{2,2}^{(2)}s_{2,2}^{(2)}=k^{\prime}\left(1-% \left(s_{1,2}^{(2)}\right)^{2}-\left(s_{2,2}^{(2)}\right)^{2}\right).$
##### 5: 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 $E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.$
##### 6: 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 $\lambda=\lambda^{m}_{m+2j+1}\left(\gamma^{2}\right)$, $j=0,1,2,\dots$. …
##### 7: 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 $A_{2p}=0$, $p=n+1,n+2,\dots$. …
##### 8: 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 …
##### 9: 34.3 Basic Properties: $\mathit{3j}$ Symbol
When any one of $j_{1},j_{2},j_{3}$ is equal to $0,\tfrac{1}{2}$, or $1$, the $\mathit{3j}$ symbol has a simple algebraic form. …For these and other results, and also cases in which any one of $j_{1},j_{2},j_{3}$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127). … Even permutations of columns of a $\mathit{3j}$ symbol leave it unchanged; odd permutations of columns produce a phase factor $(-1)^{j_{1}+j_{2}+j_{3}}$, for example, …
34.3.13 $\left((j_{1}+j_{2}+j_{3}+1)(-j_{1}+j_{2}+j_{3})\right)^{\frac{1}{2}}\begin{% pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{2}+m_{2})(j_{3}-m_{3})\right)^{\frac{% 1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-\frac{1}{2}\\ m_{1}&m_{2}-\frac{1}{2}&m_{3}+\frac{1}{2}\end{pmatrix}-\left((j_{2}-m_{2})(j_{% 3}+m_{3})\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-% \frac{1}{2}\\ m_{1}&m_{2}+\frac{1}{2}&m_{3}-\frac{1}{2}\end{pmatrix},$
34.3.15 $(2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2% })\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac{1}{2}}% \left(j_{1}^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}-1&j_{2}&j_{% 3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac{1}{% 2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}+1&j_% {2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$
##### 10: 28.14 Fourier Series
The coefficients satisfy
28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$