L’Hôpital rule for derivatives

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1: 11.4 Basic Properties

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§11.4(v) Recurrence Relations and Derivatives

… ►where ${\mathscr{H}}_{\nu }(z)$ denotes either ${\mathbf{H}}_{\nu }\left(z\right)$ or ${\mathbf{L}}_{\nu }\left(z\right)$. … ►

${\displaystyle \frac{d}{dz}}(z{\mathbf{L}}_1\left(z\right))=z{\mathbf{L}}_0\left(z\right).$

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§11.4(vi) Derivatives with Respect to Order

►For derivatives with respect to the order $\nu$, see Apelblat (1989) and Brychkov and Geddes (2005). …

2: 23.3 Differential Equations

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23.3.1 $$g_2=60\sum _{w\in 𝕃\setminus \{0\}}{w}^{-4},$$

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Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\in$: element of, $\setminus$: set subtraction and $𝕃$: lattice

A&S Ref:

18.1.1

Referenced by:

§23.3(i)

Permalink:

http://dlmf.nist.gov/23.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.3(i), §23.3 and Ch.23

… ►Given $g_2$ and $g_3$ there is a unique lattice $𝕃$ 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 $𝕃$ independently of the choice of generators. However, given any pair of generators $2{\omega }_1$, $2{\omega }_3$ of $𝕃$, and with ${\omega }_2$ defined by (23.2.1), we can identify the $e_j$ individually, via … ►

§23.3(ii) Differential Equations and Derivatives

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3: 1.4 Calculus of One Variable

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§1.4(iii) Derivatives

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Higher Derivatives

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Chain Rule

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Leibniz’s Formula

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L’Hôpital’s Rule

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4: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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primes	on function symbols: derivatives with respect to argument.

… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

5: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

►The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … … ►

§25.15(ii) Zeros

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6: 3.5 Quadrature

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§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 $q_n(x)=\frac{1}{\sqrt{h_n}}{L}_n^{(\alpha )}\left(x\right)$ the recurrence relation (3.5.30_5) takes the form …

7: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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primes	derivatives with respect to the variable, except where indicated otherwise.
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►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

8: 23.6 Relations to Other Functions

… ►In this subsection $2{\omega }_1$, $2{\omega }_3$ are any pair of generators of the lattice $𝕃$, and the lattice roots $e_1$, $e_2$, $e_3$ are given by (23.3.9). … ►

23.6.13 $$\zeta \left(u\right)=\frac{{\eta }_1}{{\omega }_1}u+\frac{\pi }{2{\omega }_1}\frac{d}{dz}\ln {\theta }_1(z,q),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.14 $$\mathrm{\wp }\left(u\right)={\left(\frac{\pi }{2{\omega }_1}\right)}^2\left(\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{3{\theta }_1^{\prime }(0,q)}-\frac{d^2}{{dz}^2}\ln {\theta }_1(z,q)\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $u$: change of variable

Referenced by:

§23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

… ►Again, in Equations (23.6.16)–(23.6.26), $2{\omega }_1,2{\omega }_3$ are any pair of generators of the lattice $𝕃$ and $e_1,e_2,e_3$ are given by (23.3.9). … ►Also, ${𝕃}_1$, ${𝕃}_2$, ${𝕃}_3$ 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. …

9: 18.9 Recurrence Relations and Derivatives

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§18.9(iii) Derivatives

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Jacobi

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Ultraspherical

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Laguerre

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Hermite

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10: 29.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by ►

$${\mathit{Ec}}_{\nu }^{2m}(z,k^2)={(-1)}^m{L}_{c\nu }^{(2m)}(\psi ,k^{\prime }{}^2),$$

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$${\mathit{Ec}}_{\nu }^{2m+1}(z,k^2)={(-1)}^m{L}_{s\nu }^{(2m+1)}(\psi ,k^{\prime }{}^2),$$

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$${\mathit{Es}}_{\nu }^{2m+1}(z,k^2)={(-1)}^m{L}_{c\nu }^{(2m+1)}(\psi ,k^{\prime }{}^2),$$

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