Recursion Formula for the Logarithmic Derivatives of Spherical Bessel Functions in the Complex Plane

The generation of numerical values for sequentially ordered logarithmic derivatives of three spherical Bessel functions by the recursion formula is considered. For a given quadrant in the right half of the complex-argument (z) plane, the forward recursion formula is found inherently convergent to one of these logarithmic derivatives, while it is found inherently convergent to a different logarithmic derivative in the opposite quadrant. The backward recursion formula converges to the third logarithmic derivative in the entire right-half place. Therefore, only two of the three functions can be calculated in any given quadrant by the recursion formula. However, the third function can be accurately approximated since it is asymptomatically equal to one or the other of the two functions that can be generated by the recursion formula, the choice depending on whether the function's order is less than or greater than an integer function of the argument, N(Z). This function, N(Z), which denotes the location of the very sharp transition zone between the regions of validity of the two asymptomatic equalities, is plotted, and from this an expression for N(Z) is developed. The approximation of the logarithmic derivative functions by a radical is discussed.