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1: 28.10 Integral Equations
§28.10(ii) Equations with Bessel-Function Kernels
2: 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 ) . …
3: 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
4: 10.68 Modulus and Phase Functions
5: 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 ) .
6: 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
7: 18.18 Sums
§18.18(vii) Poisson Kernels
Laguerre
For the modified Bessel function I ν ( z ) see §10.25(ii).
Hermite
These Poisson kernels are positive, provided that x , y are real, 0 z < 1 , and in the case of (18.18.27) x , y 0 . …
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 W
  • E. J. Weniger and J. Čížek (1990) Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Comm. 59 (3), pp. 471–493.
  • A. D. Wheelon (1968) Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.
  • C. S. Whitehead (1911) On a generalization of the functions ber x, bei x, ker x, kei x. Quart. J. Pure Appl. Math. 42, pp. 316–342.
  • M. E. Wojcicki (1961) Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.
  • E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.