^r 



ON SOME RELATIONS BETWEEN BESSEL FUNC- 

 TIONS OF THE FIRST AND OF THE SECOND 

 KIND.* 



Alexander S. Chess in. 



The general solution of Bessel's equation 



n) ^ A_l ^_L_(i_^\^ =0 



dx^ X dx \ x^j ^ 



is of the form 



(2) y = AJA'-^)-\' BK^ix), 



or of the form 



(2)' y = AJ^{x^^BJ_,{x), 



according as n is, or is not, an integral number, A and B 

 being arbitrary constants, while </„ (x) and K^ (x) denote 

 Bessel functions of the first and of the second kind, the func- 

 tions Jni^'X') and e7_„ (a;) being distinct and independent when 

 n is not an integral number. 



The differential equation (1) may be presented in a different 

 form, namely 



/i\' d'^(yVx) I 4^2 — 1\ ._ 

 ^'> ^i?- + (l 45^K»= = 0. 



or, we may say that u ^yVx is the general solution of the 

 differential equation 



(o \ Ct %C 



^> -d^2-^uf(x)=0, 



* Presented by title to The Academy of Science of St. Louis, October 

 20; 1902. 



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