limit point and limit circle boundary conditions

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►By Weyl’s alternative

n_{1}

equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for

n_{2}

. … A boundary value for the end point

a

is a linear form

\mathcal{B}

\mathcal{D}({\mathcal{L}}^{*})

of the form … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of

51

solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.

2: 1.9 Calculus of a Complex Variable

… ►

Continuity

… ►A point

z_{0}

is a limit point (limiting point or accumulation point) of a set of points

S

\mathbb{C}

(or

\mathbb{C}\cup\infty

) if every neighborhood of

z_{0}

contains a point of

S

distinct from

z_{0}

. …As a consequence, every neighborhood of a limit point of

S

contains an infinite number of points of

S

. Also, the union of

S

and its limit points is the closure of

S

. … ►A function

f(z)

is complex differentiable at a point

z

if the following limit exists: …

3: 20.13 Physical Applications

… ►In the singular limit

\Im\tau\rightarrow 0+

, the functions

\theta_{j}\left(z\middle|\tau\right)

j=1,2,3,4

, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

4: 2.1 Definitions and Elementary Properties

… ►Let

\mathbf{X}

be a point set with a limit point

c

. As

x\to c

\mathbf{X}

… ►If

c

is a finite limit point of

\mathbf{X}

, then … ►Similarly for finite limit point

c

in place of

\infty

. … ►where

c

is a finite, or infinite, limit point of

\mathbf{X}

. …

5: 33.18 Limiting Forms for Large $\ell$

§33.18 Limiting Forms for Large $\ell$

…

6: 1.4 Calculus of One Variable

… ►When this limit exists

f

is differentiable at

x

. … ►when the last limit exists. … ►If the limit exists then

f

is called Riemann integrable. … ►when this limit exists. … ►when this limit exists. …

7: 4.31 Special Values and Limits

§4.31 Special Values and Limits

►

Table 4.31.1: Hyperbolic functions: values at multiples of

\tfrac{1}{2}\pi i

► ►►

$z$	0	$\frac{1}{2}\pi\mathrm{i}$	$\pi\mathrm{i}$	$\frac{3}{2}\pi\mathrm{i}$	$\infty$
…

► ►

4.31.1

\lim_{z\to 0}\frac{\sinh z}{z}=1,

ⓘ

Symbols:: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

►

4.31.2

\lim_{z\to 0}\frac{\tanh z}{z}=1,

ⓘ

Symbols:: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

►

4.31.3

\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

8: 10.72 Mathematical Applications

… ►In regions in which (10.72.1) has a simple turning point

z_{0}

, that is,

f(z)

and

g(z)

are analytic (or with weaker conditions if

z=x

is a real variable) and

z_{0}

is a simple zero of

f(z)

, asymptotic expansions of the solutions

w

for large

u

can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order

\tfrac{1}{3}

(§9.6(i)). … ►In regions in which the function

f(z)

has a simple pole at

z=z_{0}

and

(z-z_{0})^{2}g(z)

is analytic at

z=z_{0}

(the case

\lambda=-1

in §10.72(i)), asymptotic expansions of the solutions

w

of (10.72.1) for large

u

can be constructed in terms of Bessel functions and modified Bessel functions of order

\pm\sqrt{1+4\rho}

, where

\rho

is the limiting value of

(z-z_{0})^{2}g(z)

z\to z_{0}

. …

9: 2.4 Contour Integrals

… ►

2.4.3

q(t)=\frac{1}{2\pi i}\lim\limits_{\eta\to\infty}\int_{\sigma-i\eta}^{\sigma+i% \eta}e^{tz}Q(z)\,\mathrm{d}z,

0<t<\infty

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $q(t)$ : analytic function, $Q(z)$ : Laplace transform of $q(t)$ and $\sigma$ : constant
Referenced by:: §2.4(ii)
Permalink:: http://dlmf.nist.gov/2.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §2.4(ii), §2.4 and Ch.2

… ►is seen to converge absolutely at each limit, and be independent of

\sigma\in[c,\infty)

. … ►If this integral converges uniformly at each limit for all sufficiently large

t

, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►in which

a

is finite,

b

is finite or infinite, and

\omega

is the angle of slope of

\mathscr{P}

a

, that is,

\lim(\operatorname{ph}\left(t-a\right))

t\to a

along

\mathscr{P}

. … ►The branch of

\omega_{0}=\operatorname{ph}\left(p^{\prime\prime}(t_{0})\right)

is the one satisfying

|\theta+2\omega+\omega_{0}|\leq\frac{1}{2}\pi

, where

\omega

is the limiting value of

\operatorname{ph}\left(t-t_{0}\right)

t\to t_{0}

from

b

. …

10: 1.10 Functions of a Complex Variable

… ►If

f_{2}(z)

, analytic in

D_{2}

, equals

f_{1}(z)

on an arc in

D=D_{1}\cap D_{2}

, or on just an infinite number of points with a limit point in

D

, then they are equal throughout

D

and

f_{2}(z)

is called an analytic continuation of

f_{1}(z)

. … ►An isolated singularity

z_{0}

is always removable when

\lim_{z\to z_{0}}f(z)

exists, for example

(\sin z)/z

z=0

. … ►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … ►If the path circles a branch point at

z=a

k

times in the positive sense, and returns to

z_{0}

without encircling any other branch point, then its value is denoted conventionally as

F((z_{0}-a){\mathrm{e}}^{2k\pi\mathrm{i}}+a)

. …