The Second Series Solution of the Bessel Equation: Bessel Functions of the Second Kind, Neumann Functions, Weber Functions
Definition[^1]
A second solution of the Bessel equation is called the Neumann function, denoted by $N_{\nu}(x)$ or $Y_{\nu}(x)$. For non-integer $\nu$,
$$ N_{\nu}(x)=Y_{\nu}(x)=\frac{\cos (\nu \pi)J_{\nu}(x)-J_{-\nu}(x)}{\sin (\nu\pi)} $$
For integer $\nu$, it is defined by the limit. For $n\in \mathbb{Z}$, $\nu \in \mathbb{R}\setminus\mathbb{Z}$,
$$ N_{n}(x)=\lim \limits_{\nu \rightarrow n}N_{\nu}(x) $$
Here, $J_{\pm \nu}(x)$ is the first kind Bessel function. Thus, the general solution of the Bessel equation is as follows.
$$ y(x)=AJ_{\nu}(x)+BN_{\nu}(x) $$
Here $A$, $B$ are arbitrary constants.
Explanation
$$ x^{2}y^{\prime \prime} + xy^{\prime} +(x^{2}-\nu^{2})y=0 $$
The series solution of the above Bessel equation is denoted as $J_{\pm\nu}(x)$ and is called the $\nu$th first kind Bessel function.
$$ J_{\nu}(x)=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n} }{\Gamma (n+1) \Gamma (n+\nu+1)} \left(\frac{x}{2} \right)^{2n+\nu} $$
$$ J_{-\nu}(x)=\sum \limits_{n=0}^{\infty}\frac{(-1)^{n}}{\Gamma (n+1)\Gamma (n-\nu+1)} \left( \frac{x}{2} \right)^{2n-\nu} $$
As you can see, since the two solutions are independent, the general solution is as follows.
$$ y(x)=AJ_{\nu}(x)+BJ_{-\nu}(x) $$
However, when $\nu$ is an integer, the two solutions are not linearly independent. Therefore, a second solution independent of $J_{\nu}(x)$ must be found when $\nu$ is an integer.
Consider briefly $\sin x$ and $\cos x$. The two functions are linearly independent. However, any linear combination of $\sin x$ and $\cos x$ is also linearly independent of $\sin x$. With this idea, any linear combination of $J_{\nu}(x)$ and $J_{-\nu}(x)$ is considered the second solution of the Bessel equation.
$$ \begin{equation} N_{\nu}(x)=\frac{\cos (\nu \pi)J_{\nu}(x)-J_{-\nu}(x)}{\sin (\nu\pi)} \label{eq1} \end{equation} $$
$N_{\nu}(x)$ is independent of $J_{\nu}(x)$ regardless of the condition of $\nu$. However, a problem arises again if $\nu$ is an integer, then
$$ N_{\nu}(x)=\frac{\cos (\nu \pi)J_{\nu}(x)-J_{-\nu}(x)}{\sin (\nu\pi)}=\frac{(-1)^{\nu}J_{\nu}(x)-(-1)^{\nu}J_{\nu}(x)}{0}=\frac{0}{0} $$
it becomes undefined. Therefore, when $\nu$ is an integer, it is defined using the limit as follows.
$$ N_{n}(x)=\lim \limits_{\nu \rightarrow n}N_{\nu}(x)\quad \text{for }n\in \mathbb{Z},\ \nu \in \mathbb{R}\setminus \mathbb{Z} $$
At this time, the above limit exists for arbitrary $x \ne 0$.
Theorem
For integer $\nu$, the Bessel function $J_{\pm \nu}(x)$ satisfies the following equation. In other words, it is not independent.
$$ J_{-\nu}(x)=(-1)^{\nu}J_{\nu}(x) $$
Proof
$$ J_{-\nu}(x)=\sum \limits_{n=0}^{\infty}\frac{(-1)^{n}}{\Gamma (n+1)\Gamma (n-\nu+1)} \left( \frac{x}{2} \right)^{2n-\nu} $$
By substituting $n=k+\nu$, then
$$ J_{-\nu}(x)=\sum \limits_{k=-\nu}^{\infty}\frac{(-1)^{k+\nu}}{\Gamma (k+\nu+1)\Gamma (k+1)} \left( \frac{x}{2} \right)^{2k+\nu} $$
Since the gamma function diverges for $0$ and negative integers, when $k=-\nu,-\nu+1,\cdots,-1$, the denominator’s $\Gamma (k+1)$ diverges, making $J_{-\nu}(x)=0$. Therefore,
$$ \begin{align*} J_{-\nu}(x)&=\sum \limits_{k=-\nu}^{\infty}\frac{(-1)^{k+\nu}}{\Gamma (k+\nu+1)\Gamma (k+1)} \left( \frac{x}{2} \right)^{2k+\nu} \\ &=\sum \limits_{k=0}^{\infty}\frac{(-1)^{k+\nu}}{\Gamma (k+\nu+1)\Gamma (k+1)} \left( \frac{x}{2} \right)^{2k+\nu} \\ &=(-1)^{\nu}\sum \limits_{k=0}^{\infty}\frac{(-1)^{k}}{\Gamma (k+\nu+1)\Gamma (k+1)} \left( \frac{x}{2} \right)^{2k+\nu} \\ &=(-1)^{\nu}J_{\nu}(x) \end{align*} $$
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