BesselJ[nu, ax] has as its small-argument approximation ([(ax)/2]^(nu))/Gamma[nu + 1], and has as its large-argument approximation (2/((pi)(ax)))^0.5*Cos[ax - (pi)/4 - nu(pi)/2].

BesselY[nu, ax] has as its small-argument approximation

Csc[(pi)(nu)] * ( Cos[(pi)(nu)] * ( ([(ax)/2]^(nu))/Gamma[nu + 1] ) - ([(ax)/2]^(-nu))/Gamma[1 - nu] ), and has as its large-argument approximation (2/((pi)(ax)))^0.5*Sin[ax - (pi)/4 - nu(pi)/2].

BesselJ[nu, ax] and BesselY[nu, ax] are Bessel functions of the first kind and second kind, of order nu, and having argument (ax), where a is a real number that is greater than 0, and x is a real variable.

At what value of x (as a function of a and nu) should the small-argument approximation be replaced by the large-argument approximation?

Using this x value to separate the domain into two intervals will hopefully allow BesselJ and BesselY to be closely approximated by piecewise functions that contain their small and large argument approximations.

Any help in answering this question is greatly appreciated.