L’Hôpital rule for derivatives

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1: 18.36 Miscellaneous Polynomials

… ►

§18.36(v) Non-Classical Laguerre Polynomials $L^{(-k)}_{n}\left(x\right)$ , $k=1,2\dots$

… ►For the Laguerre polynomials

L^{(\alpha)}_{n}\left(x\right)

this requires, omitting all strictly positive factors, … ►implying that, for

n\geq k

, the orthogonality of the

L^{(-k)}_{n}\left(x\right)

with respect to the Laguerre weight function

x^{-k}{\mathrm{e}}^{-x}

x\in[0,\infty)

. …These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the

L_{n}^{(-k)}(x)

polynomials, self-adjointness implying both orthogonality and completeness. … ►The resulting EOP’s,

\hat{L}^{(k)}_{n}\left(x\right)

n=1,2,\dots

satisfy …

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)

\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: 1.4 Calculus of One Variable

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§1.4(iii) Derivatives

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Higher Derivatives

… ►

Chain Rule

… ►

Leibniz’s Formula

… ►

L’Hôpital’s Rule

…

4: 23.3 Differential Equations

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23.3.1

g_{2}=60\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-4},

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Symbols:: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction and $\mathbb{L}$ : 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

\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}

\mathbb{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

…

5: 25.1 Special Notation

… ► ►►►

$k, m, n$	nonnegative integers.
…
primes	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: 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^{(\alpha)}_{n}\left(x\right)

the recurrence relation (3.5.30_5) takes the form …

7: 25.15 Dirichlet $L$ -functions

§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 … … ►

§25.15(ii) Zeros

…

8: 18.9 Recurrence Relations and Derivatives

§18.9 Recurrence Relations and Derivatives

… ►

Jacobi

… ►

Ultraspherical

… ►

Laguerre

… ►

Hermite

…

9: 23.1 Special Notation

… ► ►►►

$\mathbb{L}$	lattice in $\mathbb{C}$ .
…
primes	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)

. …

10: 23.21 Physical Applications

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23.21.1

\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,

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Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $z$ : complex and $\rho$ : roots
Permalink:: http://dlmf.nist.gov/23.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

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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},

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

… ►

23.21.3

f(\rho)=2\left((\rho-e_{1})(\rho-e_{2})(\rho-e_{3})\right)^{1/2}.

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Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

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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}}.

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\mathbb{L}$ : lattice, $u$ : change of variable, $v$ : change of variable and $w$ : change of variable
Permalink:: http://dlmf.nist.gov/23.21.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

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