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Weber–Schafheitlin discontinuous integrals

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11: 10.22 Integrals
§10.22 Integrals
Fractional Integral
WeberSchafheitlin Discontinuous Integrals, including Special Cases
This is the Weber transform. … For collections of integrals of the functions J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , and H ν ( 2 ) ( z ) , including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).
12: 10.75 Tables
  • British Association for the Advancement of Science (1937) tabulates j 0 , m , J 1 ( j 0 , m ) , j 1 , m , J 0 ( j 1 , m ) , m = 1 ( 1 ) 150 , 10D; y 0 , m , Y 1 ( y 0 , m ) , y 1 , m , Y 0 ( y 1 , m ) , m = 1 ( 1 ) 50 , 8D.

  • Wills et al. (1982) tabulates j 0 , m , j 1 , m , y 0 , m , y 1 , m for m = 1 ( 1 ) 30 , 35D.

  • Zhang and Jin (1996, p. 199) tabulates the real and imaginary parts of the first 15 conjugate pairs of complex zeros of Y 0 ( z ) , Y 1 ( z ) , Y 1 ( z ) and the corresponding values of Y 1 ( z ) , Y 0 ( z ) , Y 1 ( z ) , respectively, 10D.

  • Abramowitz and Stegun (1964, Chapter 11) tabulates 0 x J 0 ( t ) d t , 0 x Y 0 ( t ) d t , x = 0 ( .1 ) 10 , 10D; 0 x t 1 ( 1 J 0 ( t ) ) d t , x t 1 Y 0 ( t ) d t , x = 0 ( .1 ) 5 , 8D.

  • Zhang and Jin (1996, p. 270) tabulates 0 x J 0 ( t ) d t , 0 x t 1 ( 1 J 0 ( t ) ) d t , 0 x Y 0 ( t ) d t , x t 1 Y 0 ( t ) d t , x = 0 ( .1 ) 1 ( .5 ) 20 , 8D.

  • 13: 11.14 Tables
  • Abramowitz and Stegun (1964, Chapter 12) tabulates 0 x ( I 0 ( t ) 𝐋 0 ( t ) ) d t and ( 2 / π ) x t 1 𝐇 0 ( t ) d t for x = 0 ( .1 ) 5 to 5D or 7D; 0 x ( 𝐇 0 ( t ) Y 0 ( t ) ) d t ( 2 / π ) ln x , 0 x ( I 0 ( t ) 𝐋 0 ( t ) ) d t ( 2 / π ) ln x , and x t 1 ( 𝐇 0 ( t ) Y 0 ( t ) ) d t for x 1 = 0 ( .01 ) 0.2 to 6D.

  • §11.14(iv) Anger–Weber Functions
  • Bernard and Ishimaru (1962) tabulates 𝐉 ν ( x ) and 𝐄 ν ( x ) for ν = 10 ( .1 ) 10 and x = 0 ( .1 ) 10 to 5D.

  • Jahnke and Emde (1945) tabulates 𝐄 n ( x ) for n = 1 , 2 and x = 0 ( .01 ) 14.99 to 4D.

  • §11.14(v) Incomplete Functions
    14: 11.16 Software
    §11.16(iii) Integrals of Struve Functions
    §11.16(v) Anger and Weber Functions
    §11.16(vi) Integrals of Anger and Weber Functions
    15: 11.11 Asymptotic Expansions of Anger–Weber Functions
    §11.11 Asymptotic Expansions of Anger–Weber Functions
    §11.11(i) Large | z | , Fixed ν
    §11.11(ii) Large | ν | , Fixed z
    (Note that Olver’s definition of 𝐀 ν ( z ) omits the factor 1 / π in (11.10.4).) …
    16: 10.21 Zeros
    When all of their zeros are simple, the m th positive zeros of these functions are denoted by j ν , m , j ν , m , y ν , m , and y ν , m respectively, except that z = 0 is counted as the first zero of J 0 ( z ) . … Corresponding uniform approximations for y ν , m , Y ν ( y ν , m ) , y ν , m , and Y ν ( y ν , m ) , are obtained from (10.21.41)–(10.21.44) by changing the symbols j , J , Ai , Ai , a m , and a m to y , Y , Bi , Bi , b m , and b m , respectively. …
    Zeros of Y n ( n z ) and Y n ( n z )
    The first set of zeros of the principal value of Y n ( n z ) are the points z = y n , m / n , m = 1 , 2 , , on the positive real axis (§10.21(i)). … The zeros of Y n ( n z ) have a similar pattern to those of Y n ( n z ) . …
    17: 11.13 Methods of Computation
    §11.13(i) Introduction
    The treatment of Lommel and Anger–Weber functions is similar. … For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for 𝐊 ν ( z ) and 𝐌 ν ( z ) . …Other integrals that appear in §11.5(i) have highly oscillatory integrands unless z is small. … See §3.6 for implementation of these methods, and with the Weber function 𝐄 n ( x ) as an example.
    18: 10.2 Definitions
    Bessel Function of the Second Kind (Weber’s Function)
    Whether or not ν is an integer Y ν ( z ) has a branch point at z = 0 . … Except in the case of J ± n ( z ) , the principal branches of J ν ( z ) and Y ν ( z ) are two-valued and discontinuous on the cut ph z = ± π ; compare §4.2(i). Both J ν ( z ) and Y ν ( z ) are real when ν is real and ph z = 0 . For fixed z ( 0 ) each branch of Y ν ( z ) is entire in ν . …
    19: 10.15 Derivatives with Respect to Order
    10.15.5 J ν ( z ) ν | ν = 0 = π 2 Y 0 ( z ) , Y ν ( z ) ν | ν = 0 = π 2 J 0 ( z ) .
    For the notations Ci and Si see §6.2(ii). …
    10.15.9 Y ν ( x ) ν | ν = 1 2 = 2 π x ( Ci ( 2 x ) sin x ( Si ( 2 x ) π ) cos x ) .
    20: 11.15 Approximations
  • Luke (1975, pp. 416–421) gives Chebyshev-series expansions for 𝐇 n ( x ) , 𝐋 n ( x ) , 0 | x | 8 , and 𝐇 n ( x ) Y n ( x ) , x 8 , for n = 0 , 1 ; 0 x t m 𝐇 0 ( t ) d t , 0 x t m 𝐋 0 ( t ) d t , 0 | x | 8 , m = 0 , 1 and 0 x ( 𝐇 0 ( t ) Y 0 ( t ) ) d t , x t 1 ( 𝐇 0 ( t ) Y 0 ( t ) ) d t , x 8 ; the coefficients are to 20D.

  • Newman (1984) gives polynomial approximations for 𝐇 n ( x ) for n = 0 , 1 , 0 x 3 , and rational-fraction approximations for 𝐇 n ( x ) Y n ( x ) for n = 0 , 1 , x 3 . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.