About the Project

L’Hôpital rule for derivatives

AdvancedHelp

(0.003 seconds)

1—10 of 342 matching pages

1: 18.9 Recurrence Relations and Derivatives
§18.9(iii) Derivatives
Jacobi
Ultraspherical
Laguerre
Hermite
2: 11.4 Basic Properties
§11.4(v) Recurrence Relations and Derivatives
where ν ( z ) denotes either 𝐇 ν ( z ) or 𝐋 ν ( z ) . …
d d z ( z 𝐋 1 ( z ) ) = z 𝐋 0 ( z ) .
§11.4(vi) Derivatives with Respect to Order
For derivatives with respect to the order ν , see Apelblat (1989) and Brychkov and Geddes (2005). …
3: 23.3 Differential Equations
23.3.1 g 2 = 60 w 𝕃 { 0 } w 4 ,
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 ω 1 , 2 ω 3 of 𝕃 , and with ω 2 defined by (23.2.1), we can identify the e j individually, via …
§23.3(ii) Differential Equations and Derivatives
4: 1.4 Calculus of One Variable
§1.4(iii) Derivatives
Higher Derivatives
Chain Rule
Leibniz’s Formula
L’Hôpital’s Rule
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 ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
6: 25.15 Dirichlet L -functions
§25.15 Dirichlet L -functions
§25.15(i) Definitions and Basic Properties
The notation L ( s , χ ) was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … …
§25.15(ii) Zeros
7: 3.5 Quadrature
§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 ( x ) 18.3). … For the choice q n ( x ) = 1 h n L n ( α ) ( x ) the recurrence relation (3.5.30_5) takes the form …
8: 23.1 Special Notation
𝕃

lattice in .

primes

derivatives with respect to the variable, except where indicated otherwise.

The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . …
9: 23.21 Physical Applications
23.21.1 x 2 ρ e 1 + y 2 ρ e 2 + z 2 ρ e 3 = 1 ,
23.21.2 ( η ζ ) ( ζ ξ ) ( ξ η ) 2 = ( ζ η ) f ( ξ ) f ( ξ ) ξ + ( ξ ζ ) f ( η ) f ( η ) η + ( η ξ ) f ( ζ ) f ( ζ ) ζ ,
23.21.3 f ( ρ ) = 2 ( ( ρ e 1 ) ( ρ e 2 ) ( ρ e 3 ) ) 1 / 2 .
23.21.5 ( ( v ) ( w ) ) ( ( w ) ( u ) ) ( ( u ) ( v ) ) 2 = ( ( w ) ( v ) ) 2 u 2 + ( ( u ) ( w ) ) 2 v 2 + ( ( v ) ( u ) ) 2 w 2 .
10: 23.6 Relations to Other Functions
In this subsection 2 ω 1 , 2 ω 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). … Again, in Equations (23.6.16)–(23.6.26), 2 ω 1 , 2 ω 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 ( 4 K , 2 i K ) , ( 2 K 2 i K , 2 K + 2 i K ) , ( 2 K , 4 i K ) , respectively. …