Note on the Evaluation of a Certain Integral Containing Bessel's Functions

H. M. MacDonald
1909 Proceedings of the London Mathematical Society  
A PARTICULAR case of the following integral occurs in Prof. H. Lamb's paper " On the Theory of Waves propagated vertically in the Atmosphere" (p. 136 supra):- Jo and using the relation 7T 7T where yu = (c 2 -a 2 -6 2 )/2a6. * Macdonald, Proc. London Matfe. iSoc., Vol. xxxv., p. 436. 1909.] EVALUATION 0F-i CEBBA m INTEGRXL (DONaAiiNrNG BESSEL'S FUNCTIONS. 148 P 1 e~c x J n {ax) J n {bx) dx = -a~*b~*Q. Jo T a result previously obtained by the writer.* * .Again, substituting m = 0, the first
more » ... 0, the first result becomes ( J n (ax)J n (bx)K 0 (cx)xdx = Z-iir-h-^abrW-irlQl.M, Jo 3 r i that is, \ J a (ax)J n (bx) KJcx)xdx = . e-71 *, J o 2a6 sinh Vr where cosh ^ = ^ = (a 2 +& 2 +c 2 )/2a&. Substituting in the first result m = n, it becomes J n (ax)J n (bx)K n (cx)x n+l dx that is, I J n (ax) J n (bx) K n (ex) x n+1 dx Jo and therefore ( J n (ax)J n (bx)K n (cx)x n+1 dx Jo
doi:10.1112/plms/s2-7.1.142 fatcat:mk2aippcgjgwtjcnntufdhpc2u