elle Exchange moll Free 1(205-892-1862) Contact Number

SearchAdvancedHelp

(0.005 seconds)

1—10 of 853 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: 33.17 Recurrence Relations and Derivatives

… ►

33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.2 $$(\mathrm{\ell }+1)\left(1+{\mathrm{\ell }}^2ϵ\right)rh(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)h(ϵ,\mathrm{\ell };r)+\mathrm{\ell }rh(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.4 $$(\mathrm{\ell }+1)r{h}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)h(ϵ,\mathrm{\ell };r)-rh(ϵ,\mathrm{\ell }+1;r).$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

3: 33.4 Recurrence Relations and Derivatives

… ►For $\mathrm{\ell }=1,2,3,\mathrm{\dots }$, let …Then, with $X_{\mathrm{\ell }}$ denoting any of $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, or $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$, ►

33.4.2 $$R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-T_{\mathrm{\ell }}{X}_{\mathrm{\ell }}+R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1}=0,$$ $\mathrm{\ell }\ge 1$,

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $T_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.3

Permalink:

http://dlmf.nist.gov/33.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

►

33.4.3 $$X_{\mathrm{\ell }}^{\prime }=R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-S_{\mathrm{\ell }}{X}_{\mathrm{\ell }},$$ $\mathrm{\ell }\ge 1$,

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.1

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

►

33.4.4 $$X_{\mathrm{\ell }}^{\prime }=S_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }}-R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1},$$ $\mathrm{\ell }\ge 0$.

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.2

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

4: 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, ►

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

►

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

5: 33.1 Special Notation

… ► ►►

$k,\mathrm{\ell }$	nonnegative integers.
…

►The main functions treated in this chapter are first the Coulomb radial functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, $c(ϵ,\mathrm{\ell };r)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. … ►

$P_{\mathrm{\ell }}(ϵ,r)=(2\mathrm{\ell }+1)!f(ϵ,\mathrm{\ell };r)/2^{\mathrm{\ell }+1}$, $Q_{\mathrm{\ell }}(ϵ,r)=-(2\mathrm{\ell }+1)!h(ϵ,\mathrm{\ell };r)/(2^{\mathrm{\ell }+1}A(ϵ,\mathrm{\ell }))$.

►

$f^{(0)}(ϵ,\mathrm{\ell };r)=f(ϵ,\mathrm{\ell };r)$, $f(ϵ,\mathrm{\ell };r)=s(ϵ,\mathrm{\ell };r)$, $g(ϵ,\mathrm{\ell };r)=c(ϵ,\mathrm{\ell };r)$.

6: 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

…

7: 33.15 Graphics

… ►

§33.15(i) Line Graphs of the Coulomb Functions $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$

… ►

►►

Figure 33.15.2: $f(ϵ,\mathrm{\ell };r),h(ϵ,\mathrm{\ell };r)$ with $\mathrm{\ell }=1,ϵ=4$. Magnify

►

►►

Figure 33.15.3: $f(ϵ,\mathrm{\ell };r),h(ϵ,\mathrm{\ell };r)$ with $\mathrm{\ell }=0,ϵ=-1/{\nu }^2,\nu =1.5$. Magnify

►

►►

Figure 33.15.4: $f(ϵ,\mathrm{\ell };r),h(ϵ,\mathrm{\ell };r)$ with $\mathrm{\ell }=0,ϵ=-1/{\nu }^2,\nu =2$. Magnify

… ►

§33.15(ii) Surfaces of the Coulomb Functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$

…

8: 33.6 Power-Series Expansions in $\rho$

… ►where $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}=1$, $A_{\mathrm{\ell }+2}^{\mathrm{\ell }}=\eta /(\mathrm{\ell }+1)$, and ►

33.6.3 $$(k+\mathrm{\ell })(k-\mathrm{\ell }-1)A_k^{\mathrm{\ell }}=2\eta A_{k-1}^{\mathrm{\ell }}-A_{k-2}^{\mathrm{\ell }},$$ $k=\mathrm{\ell }+3,\mathrm{\ell }+4,\mathrm{\dots }$,

ⓘ

Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.6

Permalink:

http://dlmf.nist.gov/33.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►

33.6.4 $$A_k^{\mathrm{\ell }}(\eta )=\frac{{(-\mathrm{i})}^{k-\mathrm{\ell }-1}}{(k-\mathrm{\ell }-1)!}{}_2{}^{}F_1^{}(\mathrm{\ell }+1-k,\mathrm{\ell }+1-\mathrm{i}\eta ;2\mathrm{\ell }+2;2).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►where $a=1+\mathrm{\ell }\pm \mathrm{i}\eta$ and $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§5.2(i)). … ►Corresponding expansions for $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }(\eta ,\rho )$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

9: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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

… ►

$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )\sim (\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }}.$

… ►

33.5.6 $$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial, $!$: factorial (as in $n!$) and $\mathrm{\ell }$: nonnegative integer

Permalink:

http://dlmf.nist.gov/33.5.E6

Encodings:

pMML, png, TeX

See also:

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

… ►

§33.5(iv) Large $\mathrm{\ell }$

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

10: 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 $r=\mathrm{\infty }$. … ►An alternative formula for $A(ϵ,\mathrm{\ell })$ is … ►When and $\mathrm{\ell }>{(-ϵ)}^{-1/2}$ the quantity $A(ϵ,\mathrm{\ell })$ may be negative, causing $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ to become imaginary. … ►When $ϵ=-1/n^2$, $n=\mathrm{\ell }+1,\mathrm{\ell }+2,\mathrm{\dots }$, $s(ϵ,\mathrm{\ell };r)$ is $\exp \left(-r/n\right)$ times a polynomial in $r/n$, and …Note that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system. …