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11: 33.15 Graphics

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

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Figure 33.15.1: $f(ϵ,\mathrm{\ell };r),h(ϵ,\mathrm{\ell };r)$ with $\mathrm{\ell }=0,ϵ=4$. Magnify

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

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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

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

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12: About MathML

… ►, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML). … ►

Browsers supporting MathML

►The Firefox browser has traditionally had the strongest support for MathML and its native MathML is used by default. ►Recent enhancements to the WebKit engine now provide support for MathML Core. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. …

13: 28.23 Expansions in Series of Bessel Functions

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.8 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m+1}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.10 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+1)B_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.12 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+2}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

14: 33.6 Power-Series Expansions in $\rho$

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33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

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

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

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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

… ►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).

15: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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

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

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

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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

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§33.5(iv) Large $\mathrm{\ell }$

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

16: 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, …

17: 26.2 Basic Definitions

… ►Given a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell }=\mathrm{\ell }(j,k)$ such that $j={\sigma }^{\mathrm{\ell }}(k)$, where ${\sigma }^1=\sigma$ and ${\sigma }^{\mathrm{\ell }}$ is the composition of $\sigma$ with ${\sigma }^{\mathrm{\ell }-1}$. … ►The total number of partitions of $n$ is denoted by $p\left(n\right)$. …

18: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.2 $${\varphi }_n(z)={\kappa }_n{z}^n+\sum _{\mathrm{\ell }=1}^n{\kappa }_{n,n-\mathrm{\ell }}{z}^{n-\mathrm{\ell }},$$

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Symbols:

$z$: complex variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Permalink:

http://dlmf.nist.gov/18.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(ii), §18.33 and Ch.18

►where ${\kappa }_n(>0)$, and ${\kappa }_{n,n-\mathrm{\ell }}(\in ℂ)$ are constants. … ►

18.33.13 $${\varphi }_n(z)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +1\right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}{z}^{\mathrm{\ell }}=\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\lambda +1}{-\lambda -n+1};z),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(v)

Permalink:

http://dlmf.nist.gov/18.33.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

… ►Let $\mu$ be a probability measure on the unit circle of which the support is an infinite set. … ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …

19: 33.8 Continued Fractions

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33.8.1 $$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$

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Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: factor

Referenced by:

§33.8, §33.8

Permalink:

http://dlmf.nist.gov/33.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

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$a=1+\mathrm{\ell }\pm \mathrm{i}\eta ,$

… ►If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then … ►

$F_{\mathrm{\ell }}^{\prime }=u{F}_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}=q^{-1}(u-p)F_{\mathrm{\ell }},$

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20: 10.42 Zeros

… ►For example, if $\nu$ is real, then the zeros of $I_{\nu }\left(z\right)$ are all complex unless for some positive integer $\mathrm{\ell }$, in which event $I_{\nu }\left(z\right)$ has two real zeros. … ► $K_n\left(z\right)$ has no zeros in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi$; this result remains true when $n$ is replaced by any real number $\nu$. For the number of zeros of $K_{\nu }\left(z\right)$ in the sector $|\mathrm{ph}z|\le \pi$, when $\nu$ is real, see Watson (1944, pp. 511–513). …