About the Project

with Bessel-function kernels

AdvancedHelp

(0.004 seconds)

1—10 of 16 matching pages

1: 28.10 Integral Equations
§28.10(ii) Equations with Bessel-Function Kernels
2: 18.18 Sums
Laguerre
3: 10.1 Special Notation
The main functions treated in this chapter are the Bessel functions J ν ( z ) , Y ν ( z ) ; Hankel functions H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) ; modified Bessel functions I ν ( z ) , K ν ( z ) ; spherical Bessel functions 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗁 n ( 1 ) ( z ) , 𝗁 n ( 2 ) ( z ) ; modified spherical Bessel functions 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 2 ) ( z ) , 𝗄 n ( z ) ; Kelvin functions ber ν ( x ) , bei ν ( x ) , ker ν ( x ) , kei ν ( x ) . …
4: 10.62 Graphs
§10.62 Graphs
For the modulus functions M ( x ) and N ( x ) see §10.68(i) with ν = 0 . …
See accompanying text
Figure 10.62.2: ker x , kei x , ker x , kei x , 0 x 8 . Magnify
See accompanying text
Figure 10.62.3: e x / 2 ber x , e x / 2 bei x , e x / 2 M ( x ) , 0 x 8 . Magnify
See accompanying text
Figure 10.62.4: e x / 2 ker x , e x / 2 kei x , e x / 2 N ( x ) , 0 x 8 . Magnify
5: 10.68 Modulus and Phase Functions
6: 10.61 Definitions and Basic Properties
10.61.2 ker ν x + i kei ν x = e ν π i / 2 K ν ( x e π i / 4 ) = 1 2 π i H ν ( 1 ) ( x e 3 π i / 4 ) = 1 2 π i e ν π i H ν ( 2 ) ( x e π i / 4 ) .
7: 10.71 Integrals
§10.71(i) Indefinite Integrals
where M ν ( x ) and N ν ( x ) are the modulus functions introduced in §10.68(i).
§10.71(ii) Definite Integrals
§10.71(iii) Compendia
8: Bibliography B
  • J. S. Ball (2000) Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.
  • F. Bowman (1958) Introduction to Bessel Functions. Dover Publications Inc., New York.
  • T. H. Boyer (1969) Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.
  • B. L. J. Braaksma and B. Meulenbeld (1967) Integral transforms with generalized Legendre functions as kernels. Compositio Math. 18, pp. 235–287.
  • British Association for the Advancement of Science (1937) Bessel Functions. Part I: Functions of Orders Zero and Unity. Mathematical Tables, Volume 6, Cambridge University Press, Cambridge.
  • 9: 10.73 Physical Applications
    §10.73(i) Bessel and Modified Bessel Functions
    §10.73(ii) Spherical Bessel Functions
    10: Bibliography
  • M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.
  • Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.
  • Y. Ameur and J. Cronvall (2023) Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.
  • D. E. Amos (1974) Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.
  • D. E. Amos (1985) A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.