variable boundaries

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11—20 of 28 matching pages

11: 2.11 Remainder Terms; Stokes Phenomenon

… ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable

x

that is intended to be used. … ►In

\mathbb{C}

both the modulus and phase of the asymptotic variable

z

need to be taken into account. …Then numerical accuracy will disintegrate as the boundary rays

\operatorname{ph}z=\alpha

\operatorname{ph}z=\beta

are approached. In consequence, practical application needs to be confined to a sector

\alpha^{\prime}\leq\operatorname{ph}z\leq\beta^{\prime}

well within the sector of validity, and independent evaluations carried out on the boundaries for the smallest value of

|z|

intended to be used. … ►Since the ray

\operatorname{ph}z=\frac{3}{2}\pi

is well away from the new boundaries, the compound expansion (2.11.7) yields much more accurate results when

\operatorname{ph}z\to\frac{3}{2}\pi

. …

12: 10.20 Uniform Asymptotic Expansions for Large Order

… ►

§10.20(i) Real Variables

… ►

10.20.1

\left(\frac{\mathrm{d}\zeta}{\mathrm{d}z}\right)^{2}=\frac{1-z^{2}}{\zeta z^{2}}

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $\zeta(z)$ : solution
Permalink:: http://dlmf.nist.gov/10.20.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.20(i), §10.20 and Ch.10

… ►

10.20.2

\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\,\mathrm{d% }t=\ln\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\sqrt{1-z^{2}},

0<z\leq 1

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $\zeta(z)$ : solution
A&S Ref:: 9.3.38
Referenced by:: §10.20(ii)
Permalink:: http://dlmf.nist.gov/10.20.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.20(i), §10.20 and Ch.10

… ►

§10.20(ii) Complex Variables

… ►The equations of the curved boundaries

D_{1}E_{1}

and

D_{2}E_{2}

in the

\zeta

-plane are given parametrically by …

13: 1.9 Calculus of a Complex Variable

… ►

Point Sets in $\mathbb{C}$

… ►Any point whose neighborhoods always contain members and nonmembers of

D

is a boundary point of

D

. When its boundary points are added the domain is said to be closed, but unless specified otherwise a domain is assumed to be open. … ►A region is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called interior points. …

14: Mathematical Introduction

… ►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … ►Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9. This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries. … ►This means that the variable

x

ranges from 0 to 1 in intervals of 0. …

15: 3.7 Ordinary Differential Equations

… ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►The remaining two equations are supplied by boundary conditions of the form … ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …

16: 10.41 Asymptotic Expansions for Large Order

… ►

§10.41(ii) Uniform Expansions for Real Variable

… ►

10.41.7

\eta=(1+z^{2})^{\frac{1}{2}}+\ln\frac{z}{1+(1+z^{2})^{\frac{1}{2}}},

ⓘ

Defines:: $\eta$ (locally)
Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 9.7.11
Permalink:: http://dlmf.nist.gov/10.41.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(ii), §10.41 and Ch.10

►

10.41.8

p=(1+z^{2})^{-\frac{1}{2}},

ⓘ

Symbols:: $z$ : complex variable
A&S Ref:: 9.7.11
Referenced by:: §10.41(iii)
Permalink:: http://dlmf.nist.gov/10.41.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(ii), §10.41 and Ch.10

… ►

§10.41(iii) Uniform Expansions for Complex Variable

… ►The curve

E_{1}BE_{2}

in the

z

-plane is the upper boundary of the domain

\mathbf{K}

depicted in Figure 10.20.3 and rotated through an angle

-\tfrac{1}{2}\pi

. …

17: 32.11 Asymptotic Approximations for Real Variables

§32.11 Asymptotic Approximations for Real Variables

… ►with boundary condition … ►and with boundary condition … ► …

18: 11.13 Methods of Computation

… ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for

\mathbf{K}_{\nu}\left(z\right)

and

\mathbf{M}_{\nu}\left(z\right)

. … ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►For

\mathbf{M}_{\nu}\left(x\right)

both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

19: 18.39 Applications in the Physical Sciences

… ►The solutions of (18.39.8) are subject to boundary conditions at

a

and

b

. … ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►Namely the

k

th eigenfunction, listed in order of increasing eigenvalues, starting at

k=0

, has precisely

k

nodes, as real zeros of wave-functions away from boundaries are often referred to. … ►

§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

… ►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. …

20: 10.21 Zeros

… ►The parameter

t

may be regarded as a continuous variable and

\rho_{\nu}

\sigma_{\nu}

as functions

\rho_{\nu}(t)

\sigma_{\nu}(t)

t

. … ►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

\mathbf{K}

, that is, the eye-shaped domain depicted in Figure 10.20.3. … ►Lastly, there are two conjugate sets, with

n

zeros in each set, that are asymptotically close to the boundary of

\mathbf{K}

n\to\infty

. … ►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

\mathbf{K}

. … ►The only other set comprises

n

zeros that are asymptotically close to the lower boundary of

\mathbf{K}

n\to\infty

. …