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1: 10.20 Uniform Asymptotic Expansions for Large Order

… ►The curves $B{P}_1{E}_1$ and $B{P}_2{E}_2$ in the $z$-plane are the inverse maps of the line segments …

2: 10.41 Asymptotic Expansions for Large Order

… ►The curve $E_{1}B{E}_2$ in the $z$-plane is the upper boundary of the domain $𝐊$ depicted in Figure 10.20.3 and rotated through an angle $-\frac{1}{2}\pi$. …

3: 36.5 Stokes Sets

… ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►For $z=0$, the set consists of the two curves … ►

$B_{-}=-1.69916,$

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$B_{+}=0.33912.$

… ►In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). …

4: 21.7 Riemann Surfaces

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§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces

… ►Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). …Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. … ►

§21.7(iii) Frobenius’ Identity

… ►These are Riemann surfaces that may be obtained from algebraic curves of the form …

5: 28.32 Mathematical Applications

… ►Also let $ℒ$ be a curve (possibly improper) such that the quantity … ►

28.32.6 $$w(z)={\int }_{ℒ}K(z,\zeta )u(\zeta )d\zeta$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $\zeta$: variable, $u(\zeta )$: solution, $K(z,\zeta )$: solution and $ℒ$: curve

Permalink:

http://dlmf.nist.gov/28.32.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

►defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. … ►is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which $A,B$ are separation constants. Two conditions are used to determine $A,B$. …

6: 25.11 Hurwitz Zeta Function

… ►

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …The curves are almost indistinguishable for , approximately. Magnify

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25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1985a, (25), p. 231)

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally the integrand was incorrect because its numerator contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-05-08 by Clemens Heuberger

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

… ►For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►

25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Source:

Apostol (1985a, fourth equation, p. 231); with $k=1$, ${\phi }_2(x)=\frac{1}{2}({\tilde{B}}_2\left(x\right)-B_2)$

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally both integrands were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-06-27 by Gergő Nemes

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

… ►where $H_n$ are the harmonic numbers: …

7: 10.21 Zeros

… ► $B_0(\zeta )$ and $C_0(\zeta )$ are defined by (10.20.11) and (10.20.12) with $k=0$. … ►Each curve that represents an infinite string of nonreal zeros should be located on the opposite side of its straight line asymptote. … ►In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points $\pm 1$ are the boundaries of $𝐊$, that is, the eye-shaped domain depicted in Figure 10.20.3. These curves therefore intersect the imaginary axis at the points $z=\pm \mathrm{i}c$, where $c=0.66274\mathrm{\dots }$. … ►In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points $\pm 1$ is the lower boundary of $𝐊$. …

8: 18.39 Applications in the Physical Sciences

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Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applications^a

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Name of OP System	$w(x)$	$[a,b]$	Notation	Applications
Freud-Bimodal	$\exp \left(-(x^4/4-x^2/2)/\alpha \right)$	$(-\mathrm{\infty },\mathrm{\infty })$	$B_n(x)$	Fokker–Planck DVRb,c
…

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… ►

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. For $Z=+1$ the weight function, red curve, has an essential singularity at $x=-1$, as all derivatives vanish as $x\to -1+$; the green curve is ${\int }_{-1}^x{w}^{\mathrm{CP}}(y)dy$, to be compared with its histogram approximation in §18.40(ii). For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to \mathrm{\infty }$, $x_k\to -1-$. Magnify

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18.39.54 $${\mathrm{\Psi }}_{x,l}(r)=B_l(x)\sum _{n=0}^{\mathrm{\infty }}\frac{n!}{\mathrm{\Gamma }(n+2l+2)}{P}_n^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s}){\varphi }_{n,l}(sr),$$ $x=\frac{8ϵ-s^2}{8ϵ+s^2}$,

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $!$: factorial (as in $n!$), $l$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.39.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39(iv), §18.39 and Ch.18

… ►Full expressions for both $A_{x_i,l}$ and $B_l(x)$ are given in Yamani and Reinhardt (1975) and it is seen that ${|A_{x_i,l}/B_l(x_i)|}^2$ = $w_i^N/w^{\mathrm{CP}}(x_i)$ where $w_i^N$ is the Gaussian-Pollaczek quadrature weight at $x=x_i$, and $w^{\mathrm{CP}}(x_i)$ is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …