On certain integrals of Lipschitz-Hankel type involving products of bessel functions
Tóm tắt
This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;'0), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0 < b/a ^ 2, 0 ^ c/a ^ 2.
Từ khóa
Tài liệu tham khảo
Batem an H. 1944 Partial differential equations o f mathematical physics. New York: Dover Publications.
Jahnke E. & Emde F. 1945 Tables o ffunctions withformulae and curves. New York: Dover Publications.
Jeffreys H. & B. S. 1946 Methods of mathematical physics. Cam bridge University Press.
M agnus W. & O berhettinger F. 1949 Specialfunctions o f mathematical physics. New York: Chelsea Publishing Co.
N om ura Y. 1940a
1940, 6 Sci.Rep. Tohoku Univ. (1), 28, 304, Sci.Rep. Tohoku Univ. (1), 29, 22
1941, Proc. Phys.-Math. Soc. Japan (3), 23, 168
Smythe W. R. 1939 Staticand dynamic electricity. New York: M cG raw -H ill Publishing Co.
Sneddon I. N. 1951 Fourier transforms. New York: M cGraw-Hill Publishing Co.
Sura-Bura M . R. 1950 Doklady Akad. Nauk S.S.S.R (n.s.) 73 901.
Tallqvist H. 1932 Comment. Phys.-Math. Soc. Sci. Fenn 6 p art 11.
Terazaw, 1916, J.Coll. Sci. Imp. Univ. Tokyo,37, article 7.
W atson G. N. 1944 The theory of Besselfunctions 2nd ed. Cam bridge University Press.