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21: 31.1 Special Notation

… ►The main functions treated in this chapter are $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, and the polynomial ${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$. …Sometimes the parameters are suppressed.

22: 10.23 Sums

… ►where $c$ is the distance of the nearest singularity of the analytic function $f(z)$ from $z=0$, ►

10.23.11 $$a_k=\frac{1}{2\pi \mathrm{i}}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$ ,

ⓘ

Defines:

$a_k$ (locally)

Symbols:

$O_n\left(x\right)$: Neumann’s polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: nonnegative integer, $z$: complex variable and $f(t)$

A&S Ref:

9.1.83

Referenced by:

Erratum (V1.1.2) for Equation (10.23.11)

Permalink:

http://dlmf.nist.gov/10.23.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

Originally the contour of integration written incorrectly as $|z|=c^{\prime }$, has been corrected to be $|t|=c^{\prime }$.

Reported 2021-03-22 by Mark Dunster

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

…

23: 3.7 Ordinary Differential Equations

… ►

3.7.1 $$\frac{d^{2}w}{{dz}^2}+f(z)\frac{dw}{dz}+g(z)w=h(z),$$

ⓘ

Defines:

$f(z)$: function (locally), $g(z)$: function (locally) and $h(z)$: function (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $w(z)$: function

Referenced by:

§3.7(i), §3.7(ii), §3.7(ii), §3.7(ii)

Permalink:

http://dlmf.nist.gov/3.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(i), §3.7 and Ch.3

►where $f$, $g$, and $h$ are analytic functions in a domain $D\subset ℂ$. … ►

3.7.6

ⓘ

Defines:

$𝐀(\tau ,z)$: matrix (locally) and $A_{jk}(\tau ,z)$: matrix entry (locally)

Permalink:

http://dlmf.nist.gov/3.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(ii), §3.7 and Ch.3

… ►

3.7.7 $$𝐛(\tau ,z)=\left[\begin{array}{c}{b}_1(\tau ,z)\\ b_2(\tau ,z)\end{array}\right],$$

ⓘ

Defines:

$𝐛(\tau ,z)$: vector (locally) and $b_j(\tau ,z)$: vector (locally)

Permalink:

http://dlmf.nist.gov/3.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(ii), §3.7 and Ch.3

… ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …

24: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►

12.10.8 $${𝒜}_s(t)=\frac{u_s(t)}{{(t^2-1)}^{\frac{3}{2}s}},{ℬ}_s(t)=\frac{v_s(t)}{{(t^2-1)}^{\frac{3}{2}s}},$$

ⓘ

Defines:

${𝒜}_s(t)$: coefficients (locally) and ${ℬ}_s(t)$: coefficients (locally)

Symbols:

$s$: nonnegative integer, $u_s(t)$: solution and $v_s(t)$: solution

Referenced by:

§12.10(iv)

Permalink:

http://dlmf.nist.gov/12.10.E8

Encodings:

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See also:

Annotations for §12.10(ii), §12.10 and Ch.12

… ►

12.10.23 $$\eta =\frac{1}{2}\arccos t-\frac{1}{2}t\sqrt{1-t^2},$$

ⓘ

Defines:

$\eta$ (locally)

Symbols:

$\arccos z$: arccosine function

Referenced by:

§12.10(vii), §12.14(ix)

Permalink:

http://dlmf.nist.gov/12.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

… ►

12.10.32 $$\tau =\frac{1}{2}\left(\frac{t}{\sqrt{t^2-1}}-1\right),$$

ⓘ

Defines:

$\tau$ (locally)

Permalink:

http://dlmf.nist.gov/12.10.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vi), §12.10 and Ch.12

… ►

12.10.40 $$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$

ⓘ

Defines:

$\varphi (\zeta )$: function (locally)

Symbols:

$\zeta$: change of variable

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

►The function $\zeta =\zeta (t)$ is real for $t>-1$ and analytic at $t=1$. …

25: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Should $q(x)$ be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). … … ►For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. …This dilatation transformation, which does require analyticity of $q(x)$ in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of $⟨{\left(z-T\right)}^{-1}f,f⟩$. …

26: 10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.10 $$A_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{v}_j{\zeta }^{-3j/2}{U}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

ⓘ

Defines:

$A_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $v_k$: constants and $U_k(p)$: polynomial coefficient

A&S Ref:

9.3.40

Referenced by:

§10.20(i), §10.41(v), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

… ►

10.20.13 $$D_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{u}_j{\zeta }^{-3j/2}{V}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right).$$

ⓘ

Defines:

$D_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $u_k$: constants and $V_k(p)$: polynomial coefficient

A&S Ref:

9.3.46

Referenced by:

§10.20(i), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

… ►The function $\zeta =\zeta (z)$ given by (10.20.2) and (10.20.3) can be continued analytically to the $z$-plane cut along the negative real axis. … ►

10.20.18 $$c={({\tau }_0^2-1)}^{\frac{1}{2}}=0.66274\mathrm{\dots }.$$

ⓘ

Defines:

$c$ (locally)

Referenced by:

§10.41(iii)

Permalink:

http://dlmf.nist.gov/10.20.E18

Encodings:

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See also:

Annotations for §10.20(ii), §10.20 and Ch.10

… ►As $\nu \to \mathrm{\infty }$ through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for $|\mathrm{ph}z|\le \pi -\delta$, the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, being the analytic continuations of the functions defined in §10.20(i) when $\zeta$ is real. …

27: 3.3 Interpolation

… ►

3.3.3 $${\omega }_{n+1}(z)=\prod _{k=0}^n(z-z_k),$$

ⓘ

Defines:

${\omega }_{n+1}(z)$: nodal polynomial (locally)

Symbols:

$z_k$: nodes of function

Referenced by:

§3.3(iii), Erratum (V1.1.3) for Subsection 3.3(i)

Permalink:

http://dlmf.nist.gov/3.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(i), §3.3 and Ch.3

… ►

3.3.5 $$R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}{\omega }_{n+1}(x),$$

ⓘ

Defines:

$R_n(x)$: remainder (locally)

Symbols:

$!$: factorial (as in $n!$), $f(z)$: function and ${\omega }_{n+1}(z)$: nodal polynomial

A&S Ref:

25.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/3.3.E5

Encodings:

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See also:

Annotations for §3.3(i), §3.3 and Ch.3

… ►If $f$ is analytic in a simply-connected domain $D$ (§1.13(i)), then for $z\in D$, … ►

3.3.12 $$c_n=\frac{1}{(n+1)!}\max \prod _{k=n_0}^{n_1}\left|t-k\right|,$$

ⓘ

Defines:

$c_n$: bound coefficient (locally)

Symbols:

$!$: factorial (as in $n!$)

Referenced by:

§3.4(i)

Permalink:

http://dlmf.nist.gov/3.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(ii), §3.3 and Ch.3

… ►If $f$ is analytic in a simply-connected domain $D$, then for $z\in D$, …

28: 33.14 Definitions and Basic Properties

… ►

§33.14(i) Coulomb Wave Equation

… ►

§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

… ► $f(ϵ,\mathrm{\ell };r)$ is real and an analytic function of $r$ in the interval , and it is also an analytic function of $ϵ$ when . … ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

… ► $h(ϵ,\mathrm{\ell };r)$ is real and an analytic function of each of $r$ and $ϵ$ in the intervals and , except when $r=0$ or $ϵ=0$. …

29: 1.9 Calculus of a Complex Variable

… ►

Analyticity

… ►A function $f(z)$ is analytic in a domain $D$ if it is analytic at each point of $D$. A function analytic at every point of $ℂ$ is said to be entire. … ► …

30: 31.15 Stieltjes Polynomials

… ►

31.15.10 $$Q=(a_1,a_2)\times (a_2,a_3)\times \mathrm{\cdots }\times (a_{N-1},a_N),$$

ⓘ

Defines:

$Q$: set (locally)

Symbols:

$(a,b)$: open interval, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

… ►

31.15.12

ⓘ

Defines:

$\rho (z)$: weight function (locally)

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

… ►For further details and for the expansions of analytic functions in this basis see Volkmer (1999).