real case

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1: 28.29 Definitions and Basic Properties

… ►

§28.29(iii) Discriminant and Eigenvalues in the Real Case

…

2: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters

\mu

and

\nu

the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …

3: 20.1 Special Notation

… ► ►►►►

$m$ , $n$	integers.
…
$q$ $(\in\mathbb{C})$	the nome, $q=e^{i\pi\tau}$ , $0<\left\|q\right\|<1$ . Since $\tau$ is not a single-valued function of $q$ , it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$ , $\Im\tau>0$ , so that $0<q<1$ and $\tau$ and $q$ are uniquely related.
$q^{\alpha}$	$e^{i\alpha\pi\tau}$ for $\alpha\in\mathbb{R}$ (resolving issues of choice of branch).
…

…

4: 10.2 Definitions

… ►Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case

\Re\nu\geq 0

. …

5: 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. …

6: 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. An example is provided by

\operatorname{Gi}\left(x\right)

on the positive real axis. In these cases boundary-value methods need to be used instead; see §3.7(iii). …

7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Let

T

be a self-adjoint extension of differential operator

\mathcal{L}

of the form (1.18.28) and assume

T

has a complete set of

L^{2}

eigenfunctions,

\left\{\phi_{\lambda_{n}}(x)\right\}_{n=0}^{\infty}

x\in X=[a,b]

this latter being an appropriate sub-set of

\mathbb{R}

, or, in some cases

X=\mathbb{R}

itself, with real eigenvalues

\lambda_{n}

. … ►See Titchmarsh (1962a, pp. 87–90) for a first principles derivation for the case

\Re\nu\geq 1

. …

8: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►In the case of a real matrix

{\mathbf{A}}^{*}=\mathbf{A}^{\mathrm{T}}

and in the complex case

{\mathbf{A}}^{*}={\mathbf{A}}^{{\rm H}}

. …

9: 23.5 Special Lattices

… ►In this case the lattice roots

e_{1}

e_{2}

, and

e_{3}

are real and distinct. …

10: 4.2 Definitions

… ►The real and imaginary parts of

\ln z

are given by … ►The function

\exp

is an entire function of

z

, with no real or complex zeros. … ►In all other cases,

z^{a}

is a multivalued function with branch point at

z=0

. …where

\operatorname{ph}z\in[-\pi,\pi]

for the principal value of

z^{a}

, and is unrestricted in the general case. When

a

is real …