L%E2%80%99H%C3%B4pital%20rule%20for%20derivatives

SearchAdvancedHelp

(0.003 seconds)

11—20 of 467 matching pages

11: 18.5 Explicit Representations

… ►

18.5.6 $$L_n^{(\alpha )}\left(\frac{1}{x}\right)=\frac{{(-1)}^n}{n!}{x}^{n+\alpha +1}{\mathrm{e}}^{1/x}\frac{d^n}{{dx}^n}\left(x^{-\alpha -1}{\mathrm{e}}^{-1/x}\right).$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.5(ii)

Permalink:

http://dlmf.nist.gov/18.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(ii), §18.5 and Ch.18

… ►Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. … ►

$L_0\left(x\right)=1,$

… ►

$L_6\left(x\right)=\frac{1}{720}{x}^6-\frac{1}{20}{x}^5+\frac{5}{8}{x}^4-\frac{10}{3}{x}^3+\frac{15}{2}{x}^2-6x+1.$

►

$L_0^{(\alpha )}\left(x\right)=1,$

…

12: 23.3 Differential Equations

… ►

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

ⓘ

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

…

13: 23.1 Special Notation

… ► ►►►►

$𝕃$	lattice in $ℂ$.
…
primes	derivatives with respect to the variable, except where indicated otherwise.
…
$G\times H$	Cartesian product of groups $G$ and $H$, that is, the set of all pairs of elements $(g,h)$ with group operation $(g_1,h_1)+(g_2,h_2)=(g_1+g_2,h_1+h_2)$.

►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 $\eta \left(\tau \right)$. …

14: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

►

§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

…

15: 18.39 Applications in the Physical Sciences

… ►where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►with an infinite set of orthonormal $L^2$ eigenfunctions …$p$ here being the order of the Laguerre polynomial, $L_p^{(2l+1)}$ of Table 18.8.1, line 11, and $l$ the angular momentum quantum number, and where … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

16: 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 $\zeta (s,a)$, 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 $L(s,\chi )$.

17: 18.14 Inequalities

… ►

18.14.8 $${\mathrm{e}}^{-\frac{1}{2}x}\left|L_n^{(\alpha )}\left(x\right)\right|\le L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ , $\alpha \ge 0$.

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1977, Remark 4.1)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

A&S Ref:

22.14.13

Referenced by:

§18.14(i), §18.2(xii)

Permalink:

http://dlmf.nist.gov/18.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

… ►

18.14.12 $${(L_n^{(\alpha )}\left(x\right))}^2\ge L_{n-1}^{(\alpha )}\left(x\right)L_{n+1}^{(\alpha )}\left(x\right),$$ , $\alpha \ge 0$.

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.14(ii)

Permalink:

http://dlmf.nist.gov/18.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(ii), §18.14(ii), §18.14 and Ch.18

… ►Let the maxima $x_{n,m}$, $m=0,1,\mathrm{\dots },n-1$, of $|L_n^{(\alpha )}\left(x\right)|$ in $[0,\mathrm{\infty })$ be arranged so that … ►

18.14.24

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.14(iii)

Permalink:

http://dlmf.nist.gov/18.14.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

… ►The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$. …

18: 18.18 Sums

… ►

Expansion of $L^2$ functions

►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. … ►

18.18.12 $$\frac{L_n^{(\alpha )}\left(\lambda x\right)}{L_n^{(\alpha )}\left(0\right)}=\sum _{\mathrm{\ell }=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right){\lambda }^{\mathrm{\ell }}{(1-\lambda )}^{n-\mathrm{\ell }}\frac{L_{\mathrm{\ell }}^{(\alpha )}\left(x\right)}{L_{\mathrm{\ell }}^{(\alpha )}\left(0\right)}.$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.17.34_5), §18.18(iii), §18.18(iv)

Permalink:

http://dlmf.nist.gov/18.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(iii), §18.18(iii), §18.18 and Ch.18

… ►

18.18.37 $$\sum _{\mathrm{\ell }=0}^n{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right)=L_n^{(\alpha +1)}\left(x\right),$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

… ►

18.18.40 $$\sum _{\mathrm{\ell }=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right)H_{2\mathrm{\ell }}\left(x\right)H_{2n-2\mathrm{\ell }}\left(y\right)={(-1)}^n{2}^{2n}n!L_n\left(x^2+y^2\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

…

19: 18.17 Integrals

… ►

18.17.2 $${\int }_0^x{L}_m\left(y\right)L_n\left(x-y\right)dy={\int }_0^x{L}_{m+n}\left(y\right)dy=L_{m+n}\left(x\right)-L_{m+n+1}\left(x\right).$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(i)

Permalink:

http://dlmf.nist.gov/18.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

… ►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►

18.17.15 $${\mathrm{e}}^{-x}{L}_n^{(\alpha )}\left(x\right)={\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-y}{L}_n^{(\alpha +\mu )}\left(y\right)\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

►Formulas (18.17.14) and (18.17.15) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively. … ►

18.17.47 $${\int }_0^x{t}^{\alpha }\frac{L_m^{(\alpha )}\left(t\right)}{L_m^{(\alpha )}\left(0\right)}{(x-t)}^{\beta }\frac{L_n^{(\beta )}\left(x-t\right)}{L_n^{(\beta )}\left(0\right)}dt=\frac{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(\beta +1\right)}{\mathrm{\Gamma }\left(\alpha +\beta +2\right)}{x}^{\alpha +\beta +1}\frac{L_{m+n}^{(\alpha +\beta +1)}\left(x\right)}{L_{m+n}^{(\alpha +\beta +1)}\left(0\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\int$: integral, $t$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

…

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

ⓘ

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

►

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

ⓘ

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. …