It has been shown that, for any degree [mu] and integer order, v = n, the general integral (18b) fails (18a), and the generalized Bessel function [J.sup.[mu].sub.-n] that is a regular integral of the first kind of the generalized Bessel differential equation (1) must be replaced by a generalized Neumann function [Y.sup.[mu].sub.n] that is a regular integral of the second kind (Section 3.1) and hence is linearly independent, leading by linear combination of [J.sup.[mu].sub.v](z) and [Y.sup.[mu].sub.v](z) to the general integral (Section 3.2).
Generalized Neumann Function of Arbitrary Degree and Integer Order
Substitution of (2), (24), and (25) into (23) specifies explicitly the Neumann function of complex degree [mu] and integer order n:
In the case of nonzero degree, [mu] [not equal to] 0, there are alternate expressions for the three terms whose sum is (23) the generalized Neumann function: (i) the logarithmic factor multiplies the generalized Bessel function (2) that has the alternate form (4b) for nonzero degree (4a); (ii) the complementary function (25), using (3b) has the alternate form (27b) for nonzero degree (27a),
The harmonic Neumann function for the domain D is given by
But the higher-order Neumann functions are not easy to find in their explicit forms.
 Hitotumatu, Sin, On the Neumann function of a sphere, Comment.
 Nayar, B.M., Neumann function for the sphere, I, II, Indian J.
 Hazlett, R.D.; Babu, D.K., Readily computable Green's and Neumann functions for symmetry-preserving triangles, Quarterly of Applied Mathematics, 67, no.
An interior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential [N.sup.i](r, [r.sub.s]):
Likely, an exterior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential [N.sup.e](r, [r.sub.s]):
In what follows, R(r, [r.sub.s]) is also called the reflected part of the Neumann function.