# L’Hôpital rule

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##### 1: 25.15 Dirichlet $L$-functions
###### §25.15(i) Definitions and Basic Properties
The notation $L\left(s,\chi\right)$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … …
###### §3.5(i) Trapezoidal Rules
The composite trapezoidal rule is …
##### 3: 3.2 Linear Algebra
With $\mathbf{y}=[y_{1},y_{2},\dots,y_{n}]^{\rm T}$ the process of solution can then be regarded as first solving the equation $\mathbf{L}\mathbf{y}=\mathbf{b}$ for $\mathbf{y}$ (forward elimination), followed by the solution of $\mathbf{U}\mathbf{x}=\mathbf{y}$ for $\mathbf{x}$ (back substitution). … Because of rounding errors, the residual vector $\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}$ is nonzero as a rule. …
3.2.8 $\mathbf{L}=\begin{bmatrix}1&0&&&0\\ \ell_{2}&1&0&&\\ &\ddots&\ddots&\ddots&\\ &&\ell_{n-1}&1&0\\ 0&&&\ell_{n}&1\end{bmatrix},$
In the case that the orthogonality condition is replaced by $\mathbf{S}$-orthogonality, that is, $\mathbf{v}_{j}^{\rm T}\mathbf{S}\mathbf{v}_{k}=\delta_{j,k}$, $j,k=1,2,\ldots,n$, for some positive definite matrix $\mathbf{S}$ with Cholesky decomposition $\mathbf{S}=\mathbf{L}^{\rm T}\mathbf{L}$, then the details change as follows. …
$\mathbf{v}_{j+1}=\ifrac{\mathbf{L}^{-1}\left(\mathbf{L}^{-1}\right)^{\rm T}% \mathbf{u}}{\beta_{j+1}},$
##### 4: 34.5 Basic Properties: $\mathit{6j}$ Symbol
34.5.11 ${(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$
$L_{r}=l_{r}(l_{r}+1),$
Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the $\mathit{6j}$ symbol. …
##### 6: 18.4 Graphics Figure 18.4.5: Laguerre polynomials L n ⁡ ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify Figure 18.4.6: Laguerre polynomials L 3 ( α ) ⁡ ( x ) , α = 0 , 1 , 2 , 3 , 4 . Magnify Figure 18.4.8: Laguerre polynomials L 3 ( α ) ⁡ ( x ) , 0 ≤ α ≤ 3 , 0 ≤ x ≤ 10 . Magnify 3D Help Figure 18.4.9: Laguerre polynomials L 4 ( α ) ⁡ ( x ) , 0 ≤ α ≤ 3 , 0 ≤ x ≤ 10 . Magnify 3D Help
##### 7: 23.10 Addition Theorems and Other Identities
23.10.10 $\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).$
23.10.17 $\wp\left(cz|c\mathbb{L}\right)=c^{-2}\wp\left(z|\mathbb{L}\right),$
23.10.19 $\sigma\left(cz|c\mathbb{L}\right)=c\sigma\left(z|\mathbb{L}\right).$
Also, when $\mathbb{L}$ is replaced by $c\mathbb{L}$ the lattice invariants $g_{2}$ and $g_{3}$ are divided by $c^{4}$ and $c^{6}$, respectively. …
##### 8: 23.14 Integrals
23.14.1 $\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),$
23.14.2 $\int{\wp}^{2}\left(z\right)\,\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1% }{12}g_{2}z,$
23.14.3 $\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 9: 23.2 Definitions and Periodic Properties
The generators of a given lattice $\mathbb{L}$ are not unique. …where $a,b,c,d$ are integers, then $2\chi_{1}$, $2\chi_{3}$ are generators of $\mathbb{L}$ iff … When $z\notin\mathbb{L}$ the functions are related by … When it is important to display the lattice with the functions they are denoted by $\wp\left(z|\mathbb{L}\right)$, $\zeta\left(z|\mathbb{L}\right)$, and $\sigma\left(z|\mathbb{L}\right)$, respectively. … If $2\omega_{1}$, $2\omega_{3}$ is any pair of generators of $\mathbb{L}$, and $\omega_{2}$ is defined by (23.2.1), then …
##### 10: 18.41 Tables
Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_{n}\left(x\right)$, $U_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, and $0.5,1,3,5,10$ for $L_{n}\left(x\right)$ and $H_{n}\left(x\right)$. … For $P_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ see §3.5(v). …