42 BELL SYSTEM TECHNICAL JOURNAL 



For values of z<24, w^lO and v^\, they are accurately calculable 

 from the following series expansions 



C„(z\p)=2(c 1 J ll+1 (z)+c 3 J n+i (z)+c b J n+5 (z) + . .), 

 and 



S n (z;v)=4l>(c 2 Jn+2(z)+CiJ n+i (z)+C(>J n + 6 (z)+ . .), 



where the coefficients Ci, c 2 . . . are polynomials in 2v, and are inde- 

 pendent of the index n. They are 



ci = l, 



c 3 = l-(2v) 2 f 



^ = l-^(2,) 2 +^(2^-l(2,)o+(2,)«, 



c 2 = l, 

 C4 =l-(2,)«, 



^=|y-^(2,)»+(2v)* l 



9.^.4. 4.^ ft 



C8= ^r " it (2i/)2+ r< (2v)4 ~ (2y)6 ' 



The tabulation of / M (z) for values of z up to 24 and of n up to 60 

 given by Gray and Mathews and by Jahnke und Emde make the 

 computation for integral values of z rapid and precise. 



For large values of n the integrals can be accurately computed, 

 except in the neighborhood of the critical point z = n/ V 1 — v 2 , (f <1), 

 from the asymptotic formulas furnished by Gronwall. 



Without detailed computation, however, the general character of 

 the integrals can be shown as follows with an accuracy usually suffi- 

 cient for engineering purposes. By differentiation S n and C„ satisfy 

 the differential equations 



Sn = vC n , 



and 



Cn = Jn(z) — vS n , 



