Evaluation of the integral
Author:
Paul W. Schmidt
Journal:
Math. Comp. 32 (1978), 265269
MSC:
Primary 33A35
MathSciNet review:
0457812
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Abstract: Methods are developed for evaluating the integral where is the Bessel function of the first kind and order , , , and is real. Only and are included in previously published tables of integrals of Bessel functions. The integrals and are used in a technique developed by I. S. Fedorova for calculating the diameter distribution of long circular cylinders from smallangle xray, light, or neutron scattering data. The are shown to be proportional to a G function. From this result, power series expansions and recurrence relations are developed for use in evaluating the . A convenient expression is obtained for the quantity required in Fedorova's method for computing diameter distributions.
 [1]
I. S. FEDOROVA, Dokl. Akad. Nauk SSSR, v. 223, 1975, p. 1007.
 [2]
I. S. FEDOROVA & J. COLLOID, Interface Sci., v. 59, 1977, pp. 100101.
 [3]
W. MAGNUS & F. OBERHETTINGER, Formulas and Theorems for the Functions of Mathematical Physics, Chelsea, New York, 1954, p. 32.
 [4]
Yudell
L. Luke, The special functions and their approximations, Vol.
I, Mathematics in Science and Engineering, Vol. 53, Academic Press,
New YorkLondon, 1969. MR 0241700
(39 #3039)
 [5]
Ref. 4, p. 226, Eq. (7).
 [6]
Ref. 4, p. 170, Eq. (6).
 [7]
Ref. 4, p. 145, Eq. (7).
 [8]
Ref. 4, pp. 1113.
 [9]
Ref. 4, pp. 143144.
 [10]
Ref. 4, p. 229, Eq. (30).
 [11]
G.
N. Watson, A treatise on the theory of Bessel functions,
Cambridge Mathematical Library, Cambridge University Press, Cambridge,
1995. Reprint of the second (1944) edition. MR 1349110
(96i:33010)
 [12]
Ref. 11, p. 77, Eq. (12).
 [1]
 I. S. FEDOROVA, Dokl. Akad. Nauk SSSR, v. 223, 1975, p. 1007.
 [2]
 I. S. FEDOROVA & J. COLLOID, Interface Sci., v. 59, 1977, pp. 100101.
 [3]
 W. MAGNUS & F. OBERHETTINGER, Formulas and Theorems for the Functions of Mathematical Physics, Chelsea, New York, 1954, p. 32.
 [4]
 Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Sect. 5.2, Academic Press, New York and London, 1969. MR 0241700 (39:3039)
 [5]
 Ref. 4, p. 226, Eq. (7).
 [6]
 Ref. 4, p. 170, Eq. (6).
 [7]
 Ref. 4, p. 145, Eq. (7).
 [8]
 Ref. 4, pp. 1113.
 [9]
 Ref. 4, pp. 143144.
 [10]
 Ref. 4, p. 229, Eq. (30).
 [11]
 G. N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, London & New York, 1966, p. 417, Eq. (5). MR 1349110 (96i:33010)
 [12]
 Ref. 11, p. 77, Eq. (12).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804578124
PII:
S 00255718(1978)04578124
Article copyright:
© Copyright 1978
American Mathematical Society
