# L’Hôpital rule for derivatives

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##### 2: 11.4 Basic Properties
###### §11.4(v) Recurrence Relations and Derivatives
where ${\cal H}_{\nu}(z)$ denotes either $\mathbf{H}_{\nu}\left(z\right)$ or $\mathbf{L}_{\nu}\left(z\right)$. …
$\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{L}_{1}\left(z\right))=z\mathbf{L}_{0}% \left(z\right).$
###### §11.4(vi) Derivatives with Respect to Order
For derivatives with respect to the order $\nu$, see Apelblat (1989) and Brychkov and Geddes (2005). …
##### 3: 23.3 Differential Equations
Given $g_{2}$ and $g_{3}$ there is a unique lattice $\mathbb{L}$ such that (23.3.1) and (23.3.2) are satisfied. … Conversely, $g_{2}$, $g_{3}$, and the set $\{e_{1},e_{2},e_{3}\}$ are determined uniquely by the lattice $\mathbb{L}$ independently of the choice of generators. However, given any pair of generators $2\omega_{1}$, $2\omega_{3}$ of $\mathbb{L}$, and with $\omega_{2}$ defined by (23.2.1), we can identify the $e_{j}$ individually, via …
##### 5: 25.1 Special Notation
 $k,m,n$ nonnegative integers. … on function symbols: derivatives with respect to argument.
The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 6: 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 …
###### §3.5(ii) Simpson’s Rule
The $p_{n}(x)$ are the monic Laguerre polynomials $L_{n}\left(x\right)$18.3). … For the choice $q_{n}(x)=\frac{1}{\sqrt{h_{n}}}L^{(\alpha)}_{n}\left(x\right)$ the recurrence relation (3.5.30_5) takes the form …
##### 8: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … derivatives with respect to the variable, except where indicated otherwise. …
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 9: 23.21 Physical Applications
23.21.1 $\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,$
23.21.2 $(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},$
23.21.5 $\left(\wp\left(v\right)-\wp\left(w\right)\right)\left(\wp\left(w\right)-\wp% \left(u\right)\right)\left(\wp\left(u\right)-\wp\left(v\right)\right)\nabla^{2% }=\left(\wp\left(w\right)-\wp\left(v\right)\right)\frac{{\partial}^{2}}{{% \partial u}^{2}}+\left(\wp\left(u\right)-\wp\left(w\right)\right)\frac{{% \partial}^{2}}{{\partial v}^{2}}+\left(\wp\left(v\right)-\wp\left(u\right)% \right)\frac{{\partial}^{2}}{{\partial w}^{2}}.$
##### 10: 23.6 Relations to Other Functions
In this subsection $2\omega_{1}$, $2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$, and the lattice roots $e_{1}$, $e_{2}$, $e_{3}$ are given by (23.3.9). …
23.6.13 $\zeta\left(u\right)=\frac{\eta_{1}}{\omega_{1}}u+\frac{\pi}{2\omega_{1}}\frac{% \mathrm{d}}{\mathrm{d}z}\ln\theta_{1}\left(z,q\right),$
23.6.14 $\wp\left(u\right)=\left(\frac{\pi}{2\omega_{1}}\right)^{2}\left(\frac{\theta_{% 1}'''\left(0,q\right)}{3\theta_{1}'\left(0,q\right)}-\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}z}^{2}}\ln\theta_{1}\left(z,q\right)\right),$
Again, in Equations (23.6.16)–(23.6.26), $2\omega_{1},2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$ and $e_{1},e_{2},e_{3}$ are given by (23.3.9). … Also, $\mathbb{L}_{\mspace{1.0mu}1}$, $\mathbb{L}_{\mspace{1.0mu}2}$, $\mathbb{L}_{\mspace{1.0mu}3}$ are the lattices with generators $(4K,2\mathrm{i}{K^{\prime}})$, $(2K-2\mathrm{i}{K^{\prime}},2K+2\mathrm{i}{K^{\prime}})$, $(2K,4\mathrm{i}{K^{\prime}})$, respectively. …