pseudo-lemniscatic case

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11: 18.32 OP’s with Respect to Freud Weights

… ►Of special interest are the cases

Q(x)=x^{2m}

m=1,2,\dots

, and the case

Q(x)=\frac{1}{4}x^{4}-tx^{2}

(

t\in\mathbb{R}

), see §32.15. …For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case

Q(x)=x^{4}

see Bo and Wong (1999). … ►The case

Q(x)={\left|x\right|}^{\beta}

(

\beta>0

) was already introduced by Freud (1976). The special case

Q(x)=\frac{1}{4}x^{4}-tx^{2}

is of particular interest, see Clarkson and Jordaan (2018). …

12: Guide to Searching the DLMF

… ►All terms are taken to be case-insensitive, except those taken to represent math expressions (see Case Sensitivity). … ►

Case Sensitivity

►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. In the following situations, DLMF search is case-sensitive: …

13: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.

14: 6.9 Continued Fraction

… ►

6.9.1

E_{1}\left(z\right)=\cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+% \cfrac{2}{z+\cfrac{3}{1+\cfrac{3}{z+}}}}}}}\cdots,

|\operatorname{ph}z|<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{ph}$ : phase and $z$ : complex variable
A&S Ref:: 5.1.22 (special case of)
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.9 and Ch.6

…

15: 9.17 Methods of Computation

… ►However, in the case of

\operatorname{Ai}\left(z\right)

and

\operatorname{Bi}\left(z\right)

this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►In the case of

\operatorname{Ai}\left(z\right)

, for example, this means that in the sectors

\tfrac{1}{3}\pi<|\operatorname{ph}z|<\pi

we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). … ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. …In these cases boundary-value methods need to be used instead; see §3.7(iii). …

16: 12.20 Approximations

… ►As special cases of these results a Chebyshev-series expansion for

U\left(a,x\right)

valid when

\lambda\leq x<\infty

follows from (12.7.14), and Chebyshev-series expansions for

U\left(a,x\right)

and

V\left(a,x\right)

valid when

0\leq x\leq\lambda

follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …

17: 23.4 Graphics

… ►

§23.4(i) Real Variables

►Line graphs of the Weierstrass functions

\wp\left(x\right)

\zeta\left(x\right)

, and

\sigma\left(x\right)

, illustrating the lemniscatic and equianharmonic cases. … ►

See accompanying text — Figure 23.4.1: $\wp\left(x;g_{2},0\right)$ for $0\leq x\leq 9$ , $g_{2}$ = 0. …(Lemniscatic case.) Magnify

►

…

18: 25.17 Physical Applications

… ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …

19: 35.10 Methods of Computation

… ►See Yan (1992) for the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions of matrix argument in the case

m=2

, and Bingham et al. (1992) for Monte Carlo simulation on

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). …

20: 4.43 Cubic Equations

… ►

4.43.2

z^{3}+pz+q=0

ⓘ

Symbols:

\pi

: the ratio of the circumference of a circle to its diameter,

\cosh\NVar{z}

: hyperbolic cosine function,

\sinh\NVar{z}

: hyperbolic sine function,

\mathrm{i}

: imaginary unit,

\sin\NVar{z}

: sine function,

a

: real or complex constant,

z

: complex variable,

p

: real constant and

q

: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants

C

and

D

given in Eq. (4.43.1), they read

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=C$ , when $p<0$ and $C\leq 1$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=C$ , when $p<0$ and $C>1$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=D$ , when $p>0$ .

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

… ►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …