G[y_]:= ((((r^2)+(y^2)-2 r y Cg))^(-1/2))-(a/y)((((r^2)+((((a^2)/y))^2)-2 r ((a^2)/y)Cg))^(-1/2))

G'[y]/.y->a

%//Simplify

(∫_0^∞)BesselJ[0,k a]BesselJ[1,k a] (E^(-k z))\[DifferentialD]k

Simplifying identity from DLMF :

test[z_,a_]:=EllipticK[-((4 (a^2))/(z^2))]/((z/(Sqrt[(z^2)+4(a^2)]))EllipticK[(((((2a)/(Sqrt[(z^2)+4(a^2)]))))^2)])

test[9.0,3.0]

((1-kk)/(1+kk))/.kk->-((4 (a^2))/(z^2))//Simplify

(∫_0^a)BesselJ[0,k r]r\[DifferentialD]r

BesselJ[0,0]

(∫_0^∞)BesselJ[1,k a](E^(-k z))\[DifferentialD]k

Wronskian calculation

W[u1_,u2_]:=u1 D[u2,r]-D[u1,r]u2

u1=(r^L)+(L/(L+1))((a^(2L+1))/(r^(L+1)));  u2=(r^L)+(L/(L+1))((b^(2L+1))/(r^(L+1)));

(r^2)W[u1,u2]

rSqW=%//Simplify

Now simplify the radial GF:  call r1= (Subscript[r, <])   r2=(Subscript[r, >])

U1=u1/.r->r1

U2=u2/.r->r2

-U1 * U2 //Expand