Neumann functions of integer order - are they undefined by definition?

Question

I have the following definition for Neumann functions (in terms of Bessel functions)

$$ Y_n(z) = \frac{J_n(z) \cos(n\pi) - J_{-n}(z)}{\sin(n \pi)}. $$

Now my problem is concerned only with $n \in \mathbb{Z}^{\geq 0}$. So I come up with two problems with this definition.

1. $\sin(n\pi) = 0$ for all integers $n$. Now I have seen that $Y_n(z)$ is defined as it's limit as $n \rightarrow$ integer, but how does this help? That limit is still undefined, or am I missing something?

2. This second problem rests on two facts: (i) $\cos(n\pi) = (-1)^n$, and (ii) $J_{-n}(z) = (-1)^nJ_n(z)$ for integer $n$. With these two in mind it seems like for all integer $n$ and for all $z$, $Y_n(z)=0$!

I think I am going wrong somewhere very basic here but I can't seem to figure it out so I could use some help. Looking at plots of the Neumann function of real argument it is clear this is not what is going on.

Answer 1

You seem to think of $n$ in the definition of $Y_n$ as always being an integer. However, $n$ should be thought of as a real number. In that case $\sin(n \pi) \neq 0$, and similarly $\cos(n \pi) \neq (-1)^n$ (generally).

Answer 2

When $n$ is an integer there is an indeterminacy of type $0/0$. In orden to obtain $Y_n(z)$ we remove the indeterminacy by L'Hopital's rule:

$$ Y_n(z)=\lim _{\nu \rightarrow n} \frac{\frac{\partial}{\partial \nu}\left[J_\nu(z) \cos \nu \pi-J_{-\nu}(z)\right]}{\frac{\partial}{\partial \nu} \sin \nu \pi} $$ as a result, we obtain the formula $$ \begin{aligned} Y_n(z)=& \frac{2}{\pi} J_n(z)\left(\ln \frac{z}{2}+C\right)-\frac{1}{\pi} \sum_{m=0}^{n-1} \frac{(n-m-1) !}{m !}\left(\frac{z}{2}\right)^{-n+2 m} \\ &-\frac{1}{\pi} \sum_{m=0}^{\infty} \frac{(-1)^m(z / 2)^{n+2 m}}{m !(m+n) !}\left\{\sum_{k=1}^{n+m} \frac{1}{k}+\sum_{k=1}^m \frac{1}{k}\right\} \end{aligned} $$ where C is the Euler constant.