elle uppi０rt ¥ ２０５☞８９２☞１８６２ ¥ Number moll Free heine Number

SearchAdvancedHelp

(0.005 seconds)

1—10 of 629 matching pages

1: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

►As $\mathrm{\ell }\to \mathrm{\infty }$ with $ϵ$ and $r$ ($\ne 0$) fixed, ►

$f(ϵ,\mathrm{\ell };r)\sim {\displaystyle \frac{{(2r)}^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!}},$

►

$h(ϵ,\mathrm{\ell };r)\sim {\displaystyle \frac{(2\mathrm{\ell })!}{\pi {(2r)}^{\mathrm{\ell }}}}.$

3: 33.13 Complex Variable and Parameters

… ►The functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, and $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ may be extended to noninteger values of $\mathrm{\ell }$ by generalizing $(2\mathrm{\ell }+1)!=\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $U(a,b,z)$ expanded via (13.2.42). ►These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\mathrm{\ell }$. The quantities $C_{\mathrm{\ell }}\left(\eta \right)$, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$, and $R_{\mathrm{\ell }}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that ►

33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

… ►

33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

…

4: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

… ►

$$F_{\mathrm{\ell }}(0,\rho )={(\pi \rho /2)}^{1/2}{J}_{\mathrm{\ell }+\frac{1}{2}}\left(\rho \right),$$

… ►

§33.5(iv) Large $\mathrm{\ell }$

►As $\mathrm{\ell }\to \mathrm{\infty }$ with $\eta$ and $\rho$ ($\ne 0$) fixed, … ►

33.5.9 $$C_{\mathrm{\ell }}\left(\eta \right)\sim \frac{{\mathrm{e}}^{-\pi \eta /2}}{(2\mathrm{\ell }+1)!!}\sim {\mathrm{e}}^{-\pi \eta /2}\frac{{\mathrm{e}}^{\mathrm{\ell }}}{\sqrt{2}{(2\mathrm{\ell })}^{\mathrm{\ell }+1}}.$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.23(iv), §33.5(iv)

Permalink:

http://dlmf.nist.gov/33.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §33.5(iv), §33.5 and Ch.33

5: 33.14 Definitions and Basic Properties

… ►Again, there is a regular singularity at $r=0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $r=\mathrm{\infty }$. … ►The functions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ are defined by …An alternative formula for $A(ϵ,\mathrm{\ell })$ is … ►Note that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system. … ►With arguments $ϵ,\mathrm{\ell },r$ suppressed, …

6: Bibliography M

… ►

Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

ⓘ

Notes:

A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.

External Links:

Magma

Referenced by:

4th item.

Encodings:

BibTeX

… ►

ⓘ

Notes:

System for the manipulation of symbolic and numerical expressions. Includes a collection of special functions. Utilizes exact fractions, arbitrary precision integers, and arbitrary precision floating point numbers.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Precise Coulomb wave functions for a wide range of complex $\mathrm{\ell }$, $\eta$ and $z$ . Computer Physics Communications 176 (3), pp. 232–249.

ⓘ

Notes:

C++ code. Maximum accuracy 10D

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Exceptional orthogonal polynomials.

ⓘ

Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

Encodings:

BibTeX

… ►

ⓘ

Notes:

Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.

External Links:

mpmath

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

…

7: 33.16 Connection Formulas

… ►

§33.16(i) $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ in Terms of $f$ and $h$

… ►where $C_{\mathrm{\ell }}\left(\eta \right)$ is given by (33.2.5) or (33.2.6). … ►and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). … ►and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). … ►When denote $\nu$, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$, and ${\xi }_{\mathrm{\ell }}(\nu ,r)$ by (33.16.8) and (33.16.9). …

8: 33.2 Definitions and Basic Properties

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty }$ (§§2.7(i), 2.7(ii)). … ►The normalizing constant $C_{\mathrm{\ell }}\left(\eta \right)$ is always positive, and has the alternative form … ► ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$ is the Coulomb phase shift. … ► $H_{\mathrm{\ell }}^{+}(\eta ,\rho )$ and $H_{\mathrm{\ell }}^{-}(\eta ,\rho )$ are complex conjugates, and their real and imaginary parts are given by … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . …

9: 33.19 Power-Series Expansions in $r$

… ►

${\alpha }_0=2^{\mathrm{\ell }+1}/(2\mathrm{\ell }+1)!,$

… ►

33.19.3 $$2\pi h(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{2\mathrm{\ell }}\frac{(2\mathrm{\ell }-k)!{\gamma }_k}{k!}{(2r)}^{k-\mathrm{\ell }}-\sum _{k=0}^{\mathrm{\infty }}{\delta }_k{r}^{k+\mathrm{\ell }+1}-A(ϵ,\mathrm{\ell })\left(2\ln |2r/\kappa |+\mathrm{\Re }\psi \left(\mathrm{\ell }+1+\kappa \right)+\mathrm{\Re }\psi \left(-\mathrm{\ell }+\kappa \right)\right)f(ϵ,\mathrm{\ell };r),$$ $r\ne 0$.

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity, $A(ϵ,\mathrm{\ell })$: function, ${\gamma }_k$: term and ${\delta }_k$: term

Referenced by:

§33.19

Permalink:

http://dlmf.nist.gov/33.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

►Here $\kappa$ is defined by (33.14.6), $A(ϵ,\mathrm{\ell })$ is defined by (33.14.11) or (33.14.12), ${\gamma }_0=1$, ${\gamma }_1=1$, and … ►

${\delta }_0=\left({\beta }_{2\mathrm{\ell }+1}-2(\psi \left(2\mathrm{\ell }+2\right)+\psi \left(1\right))A(ϵ,\mathrm{\ell })\right){\alpha }_0,$

►

${\delta }_1=\left({\beta }_{2\mathrm{\ell }+2}-2(\psi \left(2\mathrm{\ell }+3\right)+\psi \left(2\right))A(ϵ,\mathrm{\ell })\right){\alpha }_1,$

…

10: 24.16 Generalizations

… ►For $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, Bernoulli and Euler polynomials of order $\mathrm{\ell }$ are defined respectively by …When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\mathrm{\ell }$ : …Also for $\mathrm{\ell }=1,2,3,\mathrm{\dots }$, … ►For extensions of $B_n^{(\mathrm{\ell })}\left(x\right)$ to complex values of $x$, $n$, and $\mathrm{\ell }$, and also for uniform asymptotic expansions for large $x$ and large $n$, see Temme (1995b) and López and Temme (1999b, 2010b). … ►(This notation is consistent with (24.16.3) when $x=\mathrm{\ell }$.) …