limiting forms as order tends to integers

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11: 10.52 Limiting Forms

§10.52 Limiting Forms

►

§10.52(i) $z\to 0$

… ►

§10.52(ii) $z\to\infty$

… ►

10.52.5

{\mathsf{i}^{(1)}_{n}}\left(z\right)\sim{\mathsf{i}^{(2)}_{n}}\left(z\right)% \sim\tfrac{1}{2}z^{-1}e^{z},

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $z$ : complex variable and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/10.52.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §10.52(ii), §10.52 and Ch.10

►

10.52.6

\mathsf{k}_{n}\left(z\right)\sim\tfrac{1}{2}\pi z^{-1}e^{-z}.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.52.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §10.52(ii), §10.52 and Ch.10

12: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

►

§33.10(i) Large $\rho$

►As

\rho\to\infty

with

\eta

fixed, … ►

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

…

13: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►Higher-order determinants are natural generalizations. …An

n

th-order determinant expanded by its

j

th row is given by … ►If

D_{n}[a_{j,k}]

tends to a limit

L

n\to\infty

, then we say that the infinite determinant

D_{\infty}[a_{j,k}]

converges and

D_{\infty}[a_{j,k}]=L

. … ►The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. ►Let the columns of matrix

\mathbf{S}

be these eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

, then

{\mathbf{S}}^{-1}={\mathbf{S}}^{{\rm H}}

, and the similarity transformation (1.2.73) is now of the form

{\mathbf{S}}^{{\rm H}}\mathbf{A}\mathbf{S}=\lambda_{i}\delta_{i,j}

. …

14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►where the limit has to be understood in the sense of

L^{2}

convergence in the mean: … ►Let

T

be the self adjoint extension of a formally self-adjoint differential operator

\mathcal{L}

of the form (1.18.28) on an unbounded interval

X\subset\mathbb{R}

, which we will take as

X=[0,+\infty)

, and assume that

q(x)\to 0

monotonically as

x\to\infty

, and that the eigenfunctions are non-vanishing but bounded in this same limit. … ► A boundary value for the end point

a

is a linear form

\mathcal{B}

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

of the form …where

\alpha

and

\beta

are given functions on

X

, and where the limit has to exist for all

f

. … ►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. …

15: 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}

. …

16: 18.11 Relations to Other Functions

… ►

§18.11(ii) Formulas of Mehler–Heine Type

►

Jacobi

… ►

Laguerre

… ►

Hermite

… ►The limits (18.11.5)–(18.11.8) hold uniformly for

z

in any bounded subset of

\mathbb{C}

17: 33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

►

§33.5(i) Small $\rho$

►As

\rho\to 0

with

\eta

fixed, … ►

§33.5(iii) Small $|\eta|$

… ►

§33.5(iv) Large $\ell$

…

19: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. The canonical form of differential equation for these problems is given by … ►If

f(z)

has a double zero

z_{0}

, or more generally

z_{0}

is a zero of order

m

m=2,3,4,\dotsc

, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order

1/(m+2)

. … ►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}

. … ►Then for large

u

asymptotic approximations of the solutions

w

can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on

u

and

\alpha

). …

20: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function

f(n)

for large

n

is often difficult to determine because the function values can fluctuate considerably as

n

increases. It is more fruitful to study partial sums and seek asymptotic formulas of the form …where

F(x)

is a known function of

x

, and

O\left(g(x)\right)

represents the error, a function of smaller order than

F(x)

for all

x

in some prescribed range. … ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. … ►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). …

limiting forms as order tends to integers

11—20 of 929 matching pages

11: 10.52 Limiting Forms

§10.52 Limiting Forms

§10.52(i) $z\to 0$

§10.52(ii) $z\to\infty$

12: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10(i) Large $\rho$

§33.10(ii) Large Positive $\eta$

§33.10(iii) Large Negative $\eta$

13: 1.3 Determinants, Linear Operators, and Spectral Expansions

14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

15: 2.1 Definitions and Elementary Properties

16: 18.11 Relations to Other Functions

§18.11(ii) Formulas of Mehler–Heine Type

Jacobi

Laguerre

Hermite

17: 33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5(i) Small $\rho$

§33.5(iii) Small $|\eta|$

§33.5(iv) Large $\ell$

18: 10.30 Limiting Forms

§10.30 Limiting Forms

§10.30(i) $z\to 0$

§10.30(ii) $z\to\infty$

19: 10.72 Mathematical Applications

20: 27.11 Asymptotic Formulas: Partial Sums

limiting forms as order tends to integers

11—20 of 929 matching pages

11: 10.52 Limiting Forms

§10.52 Limiting Forms

§10.52(i) z → 0

§10.52(ii) z → ∞

12: 33.10 Limiting Forms for Large ρ or Large | η |

§33.10 Limiting Forms for Large ρ or Large | η |

§33.10(i) Large ρ

§33.10(ii) Large Positive η

§33.10(iii) Large Negative η

13: 1.3 Determinants, Linear Operators, and Spectral Expansions

14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

15: 2.1 Definitions and Elementary Properties

16: 18.11 Relations to Other Functions

§18.11(ii) Formulas of Mehler–Heine Type

Jacobi

Laguerre

Hermite

17: 33.5 Limiting Forms for Small ρ , Small | η | , or Large ℓ

§33.5 Limiting Forms for Small ρ , Small | η | , or Large ℓ

§33.5(i) Small ρ

§33.5(iii) Small | η |

§33.5(iv) Large ℓ

18: 10.30 Limiting Forms

§10.30 Limiting Forms

§10.30(i) z → 0

§10.30(ii) z → ∞

19: 10.72 Mathematical Applications

20: 27.11 Asymptotic Formulas: Partial Sums

§10.52(i) $z\to 0$

§10.52(ii) $z\to\infty$

12: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10(i) Large $\rho$

§33.10(ii) Large Positive $\eta$

§33.10(iii) Large Negative $\eta$

17: 33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5(i) Small $\rho$

§33.5(iii) Small $|\eta|$

§33.5(iv) Large $\ell$

§10.30(i) $z\to 0$

§10.30(ii) $z\to\infty$