The solution corresponding to the larger root cr 1 = p can be obtained first. Excluding the case of p = -1/2, then:

a1 = a3 = a5 = ... 0

a a

and, by induction:

oo x2m+p YI(x)=aoxP +ao _L(-1j 22mm!(p+ 1Xp+ 2) ... (p+m)

The solution corresponding to the smaller root cr2 = -p can be obtained by substituting -p for +pin eq. (3.3) resulting in:

If p i-integer, then: Yh = ciJp(x)+c2Lp(x)


The expression for the Wronskian can be obtained from the form given in eq. (1.28):

Thus: Lim x W(x)~ W0 x~O


To calculate W 0, it is necessary to account for the leading terms only, since the form of W - 1/x. Thus:

1, _ rz fr2J-I P r(p+ I) I ~ fr2Jp-J

(Appendix B1)

then, the Wronskian is given by:


Another solution that also satisfies eq. (3.1), first introduced by Weber, takes the form: Y (X) -- cosp1r Jp.{x)-Lp{x)

P smp1r p -:f. integer (3.6)

such that the general solution can be written in the form known as Weber function: y = c1 Jp{x) +c2 Yp{x) p -:f. integer

Using the linear transformation formula, the Wronskian becomes:

w(JP, Yp)= det[aij]w(JP,J-P) as given by eq. ( 1.13 ), where:

a 21 = cotp1r

so that:

a22 = -Vsin p1r



To obtain the second solution, the methods developed in Section (2.4) are applied. From the recurrence formula, eq. (3.2), one obtains the following by setting p = 0:

Again, by induction, one can show that the even indexed coefficients are:


00 x2m+u y(x,u)=aoxu +ao I(-l)m ( j( )2 ( )2

y 2(x) = 8u = a0x logx+a 0 logx L ( j( )2 ( )2 u -0 u+2 u+4 ... u+2m o-m =1

which results in the second solution y2 as:



m=O To obtain the second solution for cr2 = -n, it is necessary to check a2n( -n) for boundedness. Substituting p = n in the recurrence formula (3.2) gives:

am+2 = (m + 2 + u - n Xm + 2 + u + n) a

m=O, 1,2, ...