Hankel expansions

(0.002 seconds)

21—30 of 34 matching pages

21: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►For the functions

J_{2\mu}

Y_{2\mu}

{H^{(1)}_{2\mu}}

, and

{H^{(2)}_{2\mu}}

see §10.2(ii), and for the

\mathrm{env}

functions associated with

J_{2\mu}

and

Y_{2\mu}

see §2.8(iv). … ►This reference also includes error bounds and extensions to asymptotic expansions and complex values of

x

. … ►This reference also includes error bounds and extensions to asymptotic expansions and complex values of

x

. ►

§13.21(iv) Large $\kappa$ , Other Expansions

… ►For asymptotic expansions having double asymptotic properties see Skovgaard (1966). …

22: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►For asymptotic expansions of

\phi\left(\rho,\beta;z\right)

z\to\infty

in various sectors of the complex

z

-plane for fixed real values of

\rho

and fixed real or complex values of

\beta

, see Wright (1935) when

\rho>0

, and Wright (1940b) when

-1<\rho<0

. For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when

\rho>0

, and Wong and Zhao (1999a) when

-1<\rho<0

. … ►This reference includes exponentially-improved asymptotic expansions for

E_{a,b}\left(z\right)

when

|z|\to\infty

, together with a smooth interpretation of Stokes phenomena. … ►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).

23: 28.23 Expansions in Series of Bessel Functions

§28.23 Expansions in Series of Bessel Functions

… ►

{\cal C}_{\mu}^{(3)}={H^{(1)}_{\mu}},

►

{\cal C}_{\mu}^{(4)}={H^{(2)}_{\mu}};

… ►

24: 10.18 Modulus and Phase Functions

… ►

10.18.1

M_{\nu}\left(x\right)e^{i\theta_{\nu}\left(x\right)}={H^{(1)}_{\nu}}\left(x% \right),

ⓘ

Defines:: $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions
Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.18(i), §10.18 and Ch.10

►

10.18.2

N_{\nu}\left(x\right)e^{i\phi_{\nu}\left(x\right)}={H^{(1)}_{\nu}}'\left(x% \right),

ⓘ

Defines:: $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions
Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.18(i), §10.18 and Ch.10

… ►

§10.18(iii) Asymptotic Expansions for Large Argument

… ►the general term in this expansion being … ►In (10.18.17) and (10.18.18) the remainder after

n

terms does not exceed the

(n+1)

th term in absolute value and is of the same sign, provided that

n>\nu-\frac{1}{2}

for (10.18.17) and

-\frac{3}{2}\leq\nu\leq\frac{3}{2}

for (10.18.18).

25: 9.17 Methods of Computation

… ►

§9.17(i) Maclaurin Expansions

… ►For large

|z|

the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. Since these expansions diverge, the accuracy they yield is limited by the magnitude of

|z|

. … ►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to

\operatorname{Ai}\left(z\right)

\operatorname{Bi}\left(z\right)

, and their derivatives. … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

26: Bibliography S

… ►

A. Salem (2013) Some properties and expansions associated with the

q

-digamma function. Quaest. Math. 36 (1), pp. 67–77.

ⓘ

External Links:: ISSN 1607-3606, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §5.18(ii), §5.18(ii).
Encodings:: BibTeX

… ►

J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

ⓘ

External Links:: FullText, Document, MathReview (L. Fox), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

J. D. Secada (1999) Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

ⓘ

External Links:: ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

A. Sharples (1971) Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

ⓘ

External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

ⓘ

External Links:: Abstract
Referenced by:: §18.16(v).
Encodings:: BibTeX

…

27: Bibliography H

… ►

P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

ⓘ

External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

ⓘ

External Links:: Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(iii), §2.5(ii).
Encodings:: BibTeX

… ►

E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

ⓘ

External Links:: ISSN 0096-3518, FullText, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

Harvard University (1945) Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives. Harvard University Press, Cambridge, MA.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

H. W. Hethcote (1970) Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

…

28: 10.75 Tables

… ►

§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

… ►

•

Döring (1966) tabulates all zeros of $Y_{0}\left(z\right)$ , $Y_{1}\left(z\right)$ , ${H^{(1)}_{0}}\left(z\right)$ , ${H^{(1)}_{1}}\left(z\right)$ , that lie in the sector $|z|<158$ , $|\operatorname{ph}z|\leq\pi$ , to 10D. Some of the smaller zeros of $Y_{n}\left(z\right)$ and ${H^{(1)}_{n}}\left(z\right)$ for $n=2,3,4,5,15$ are also included.

… ►

•

Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of ${H^{(1)}_{0}}\left(z\right)$ and ${H^{(1)}_{1}}\left(z\right)$ , 8D.

►

•

Kerimov and Skorokhodov (1987) tabulates 100 complex double zeros $\nu$ of $Y_{\nu}'\left(ze^{-\pi i}\right)$ and ${H^{(1)}_{\nu}}'\left(ze^{-\pi i}\right)$ , 8D.

… ►

•

Olver (1960) tabulates $a^{\prime}_{n,m}$ , $\mathsf{j}_{n}\left(a^{\prime}_{n,m}\right)$ , $b^{\prime}_{n,m}$ , $\mathsf{y}_{n}\left(b^{\prime}_{n,m}\right)$ , $n=1(1)20$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

…

29: Bibliography B

… ►

C. B. Balogh (1967) Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

ⓘ

External Links:: ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.45.
Encodings:: BibTeX

… ►

R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

ⓘ

External Links:: ISSN 0893-9659, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

B. C. Berndt and R. J. Evans (1984) Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

ⓘ

External Links:: ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §2.10(iii).
Encodings:: BibTeX

… ►

R. Bo and R. Wong (1994) Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

… ►

J. Brüning (1984) On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

ⓘ

External Links:: ISSN 0003-889X, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(ii).
Encodings:: BibTeX

…

30: Bibliography L

… ►

C. Leubner and H. Ritsch (1986) A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §2.4(vi).
Encodings:: BibTeX

… ►

X. Li and R. Wong (1994) Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (Walter Schempp), ZentralBlatt
Referenced by:: §2.6(iii).
Encodings:: BibTeX

… ►

J. L. López (2001) Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt
Referenced by:: §19.27(vi).
Encodings:: BibTeX

… ►

T. A. Lowdon (1970) Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

ⓘ

External Links:: Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §12.12, §12.16.
Encodings:: BibTeX

… ►

D. Ludwig (1966) Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.

ⓘ

External Links:: Document, MathReview (Colin Clark), ZentralBlatt
Referenced by:: §36.12(iii), §36.14(i).
Encodings:: BibTeX

…