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4-... m=0 :n ’(o + 2 - n) ... (o + 2m- n): ~-’+"~+ n) ... (o + 2m r mn) + ao logx ~ L! (~ + ---- ~=-n ] Thus, the solution corresponding to the second root ~ = -n becomes: ~ ---- -n 2m-n x CHAPTER 3 48 Y2(X) = -’~ m=O ] oo (n - m- 1)! + log x Jn(x)+ ~ g(n - 1) (X/2)2m+n [g(m)+ g(m + -~" ~ (-1)m m! (m + m=0 where -n+l ao 2 (n - was set equal to one. Thesecondsolution includes the first solution given in eq. ~-n "--’ m=0 (x~ (nm! 1 ~ (_l)m (x//2)2m+" 2 m! (m + n)! m=0 m,1), [g(m)+g(m+n)] Define: Yn(x) = ~[(~’- log2)JnCx) = ~f ~’ + log(~)] [ _1 ~(_l)m 2 m=0 Y2Cx)] J,~(x)--~ m = n-1 (x~)2m-" (n - m- 1) m=O (~)2m+.

M=0 m,1), [g(m)+g(m+n)] Define: Yn(x) = ~[(~’- log2)JnCx) = ~f ~’ + log(~)] [ _1 ~(_l)m 2 m=0 Y2Cx)] J,~(x)--~ m = n-1 (x~)2m-" (n - m- 1) m=O (~)2m+. (m+n)! 11) where Yn(X) is knownas the Bessei function of the second kind of order n, the Neumannfunetlon of order n. Thus. the solutions for p = n is: if p = n = integer Yh = ClJn(x) + C2Yn(x) The solutions of eq. 1) are also knownas Cylindrical Bessel functions. 6). 6) results in an indeterminate function. Thus: Y,(x)= Lim c°sPnJl’(x)-J-l’(x) sin pn p -~ n SPECIAL 49 FUNCTIONS =-~Zsin P~JP (x) + cos P/t ~/¢9pJP(X)-/~/cgPJ-P ffcospTr = ;{~ Jp(x)-(-1)~ J_p(X)}p !

The first part of the solution with ao maybe a finite polynomialor an infinite series, dependingon the order of the recurrence formula and on the integer k. If gk(O2)does not vanish, then one must find another methodto obtain the second solution. A new solution similar to Case (b) is developednext by removingthe constant o - o~ from the demoninatorof an(o). Since the characteristic equation in eq. ~)(0. - ¢rz) then multiplying eq. :~) = ao x~’-2 0. 1 0. (0")x n=0 n=0 n=k so that the coefficient ak is the first term of the secondseries.