normal values

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1: 28.7 Analytic Continuation of Eigenvalues

… ►The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of

q

are normal values. …

2: 28.19 Expansions in Series of $\mathrm{me}_{\nu+2n}$ Functions

… ►Let

q

be a normal value (§28.12(i)) with respect to

\nu

, and

f(z)

be a function that is analytic on a doubly-infinite open strip

S

that contains the real axis. …

3: 28.12 Definitions and Basic Properties

… ►

28.12.2

\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.12(i), §28.12 and Ch.28

►As in §28.7 values of

q

for which (28.2.16) has simple roots

\lambda

are called normal values with respect to

\nu

. For real values of

\nu

and

q

all the

\lambda_{\nu}\left(q\right)

are real, and

q

is normal. … ►If

q

is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of

z

and

q

by the normalization …

4: 28.11 Expansions in Series of Mathieu Functions

… ►Let

f(z)

be a

2\pi

-periodic function that is analytic in an open doubly-infinite strip

S

that contains the real axis, and

q

be a normal value (§28.7). …

5: 19.31 Probability Distributions

… ►

R_{G}\left(x,y,z\right)

and

R_{F}\left(x,y,z\right)

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. …

6: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions

\gamma\left(a,z\right)

and

\Gamma\left(a,z\right)

are defined by …Except where indicated otherwise in the DLMF these principal values are assumed. … ►Normalized functions are: … ►In this subsection the functions

\gamma

and

\Gamma

have their general values. … ►(8.2.9) also holds when

a

is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …

7: 3.6 Linear Difference Equations

… ►The process is then repeated with a higher value of

N

, and the normalized solutions compared. … ►The normalizing factor

\Lambda

can be the true value of

w_{0}

divided by its trial value, or

\Lambda

can be chosen to satisfy a known property of the wanted solution of the form …

8: 31.9 Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. ►The normalization constant

\theta_{m}

is given by ►

31.9.3

\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},

ⓘ

Defines:: $\theta_{m}$ : normalization constant (locally)
Symbols:: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\partial\NVar{x}$ : partial differential of $x$ , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\zeta$ : point and $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients
Permalink:: http://dlmf.nist.gov/31.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

… ►The right-hand side may be evaluated at any convenient value, or limiting value, of

\zeta

(0,1)

since it is independent of

\zeta

. … ►For further information, including normalization constants, see Sleeman (1966a). …

9: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

… ►

§1.6(iii) Vector-Valued Functions

… ►An orientable surface is oriented if suitable normals have been chosen. … ►

Stokes’s Theorem

… ►

10: 28.2 Definitions and Basic Properties

… ►For nonnegative real values of

q

, see Figure 28.2.1. … ►

Distribution

… ►

Change of Sign of $q$

… ►For simple roots

q

of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations … …