Bessel's Equation

$$x^2y'' + xy' + (x^2- p^2)y = 0, \tag{34}$$

where $p \in R \geq 0$ , is known as Bessel's differential equation of order $p$

Here, $P (x) = \frac{1}{x}$ and $Q (x) = \frac{x^{2} - p^{2}}{x^{2}}$ are not analytic, but $x P (x)$ and $x^{2} Q (x)$ are analytic around $x_{0} = 0$
Hence, $x = 0$ is a regular singular point.
Therefore, we can get a Frobenius solution, where $p_{0} = x P (x) |_{x = 0} = 1$ and $q_{0} = x^{2} Q (x) |_{x = 0} = - p^{2}$
Hence, indicial equation takes the form as $$r^2 - p^2 = 0$$with solutions $m_{1, 2} = \pm p$ . Since $m_{1} \geq m_{2}$ ,
if $m_{1} - m_{2} = 2 p$ is neither zero nor an integer, thus one solution is $$y = x^p \sum_{m = 0}^\infty a_{m}x^m = \sum a_{m}x^{m+p}; a_{0} \neq 0$$

Substituting the solution in $(34)$ , we get $$\begin{align} a_{1} &= 0 \ a_{m} &= -\frac{a_{m-2}}{m(2p + m)},\ m \geq 2 \end{align}$$
And for different $m$ , we get $$\begin{align} m = 2& \ a_{2} &= -\frac{a_{0}}{2^21!(p+1)} \ \ m=4& \ a_{4} &= (-1)^2 \frac{a_{0}}{2^4 2!(p+1)(p+2)}\end{align}$$
For $m = 3, 5, 7 \dots$ we obtain $a_{m} = 0$
Extrapolating, one solution of Bessel's equation is $$y_{1} = a_{0}x^p \sum_{n=0}^\infty (-1)^n \frac{\left( \frac{x}{2} \right)^{2n}}{n!(p+1)(p+2)\dots(p+n)} \tag{39}$$

Bessel function of the first kind

The solution $y_{1}$ is called Bessel Function of first kind of order $p$ for $a_{0} = \frac{1}{2^{p} p!}$ and is denoted by $J_{p}$
Hence, $$J_{p} = \frac{1}{2^pp!}x^p \sum_{n=0}^\infty (-1)^n \frac{\left( \frac{x}{2} \right)^{2n}}{n!(p+1)(p+2)\dots(p+n)}$$
Combining the $2^{p} p!$ into the summation, we obtain $$J_{p} = \sum (-1)^n \frac{(x/2)^{2n+p}}{n!(p+n)!} \tag{41}$$
For $p = 0$ , $$J_{0} = 1- \frac{x^2}{2^2} + \frac{x^4}{2^24^2} - \frac{x^6}{2^24^26^2} \dots$$

Important

The term $(p + n)!$ is only defined for integral $p$ .
When $p$ is not an integer, we take $$a_{0} = \frac{1}{2^p\cdot\Gamma(p+1)}$$and in this case, $$J_{p} = \sum_{n=0}^\infty (-1)^n \frac{(x/2)^{2n+p}}{\Gamma(n+1)\Gamma(p+n+1)}$$

Theorem on

Γ

function

$Γ (n) = \int_{0}^{\infty} t^{n - 1} e^{- t} \cdot d t, n > 0$

Hence $Γ (n + 1) = n Γ (n)$ and $Γ (1) = 1$

Therefore for non-negative integral $n$ , $Γ (n + 1) = n!$
The recurrence relation is not defined for $n < 0$
But if we write $$\Gamma(n) = \frac{\Gamma(n+1)}{n}$$then we can see RHS is defined for $0 < n + 1 < 1$ or $- 1 < n < 0$ , ie for fractional negative $n$
Extrapolating, we can say $Γ (n)$ is defined for all $n \in (- 1, \infty)$ . When $n < - 1$ , then $Γ (n) \to \pm \infty$

Also $$\Gamma\left( \frac{1}{2} \right) = \sqrt{ \pi }$$

Second solution of Bessel's equation

When $m_{1} - m_{2} = 2 p$ is neither zero not an integer, as we had taken above, the other solution must be $J_{- p}$
Substituting directly in $(41)$ , we get $$J_{-p} = \sum_{n=0}^\infty (-1)^n \frac{(x/2)^{2n-p}}{\Gamma(n+1) \Gamma(n-p+1)}$$
Notice that this solution has unbounded first term $$\frac{1}{\Gamma(-p+1)} \left( \frac{x}{2} \right)^{-p}$$
This term shoots to infinity as $x \to 0$ , which is not the same as $J_{p}$ which remains $0$ . Thus the terms $J_{p}$ and $J_{- p}$ must be LI, allowing us to model the general solution $$y = c_{1}J_{p}(x) + c_{2}J_{-p}(x) \tag{45}$$
But when $p$ is an integer, say some $m \geq 0$ , then we can easily show that $J_{- m} = (- 1)^{m} J_{m}$ , thus, they cannot be LI.
To proceed, we define $$Y_{p}(x) = \frac{J_{p}(x)\cos p\pi - J_{-p}(x)}{\sin p\pi}$$
$Y_{p}$ , called the special Bessel function of the second kind is a specific form of $(45)$ with particular choice of $c_{1}, c_{2}$ .
Thus general solution is of the form (equivalent to $(45)$ ) $$y = c_{1}J_{p}(x) + c_{2}Y_{p}(x),\ \ \ \ \ \ p \text{ is not an integer}$$
And for the case when $p$ is an integer $m$ , $$Y_{m}(x) = \lim_{ p \to \infty } Y_{p}(x)$$
Hence, $$y = c_{1}J_{p}(x) + c_{2}Y_{p}(x)$$represents general solution of the Bessel's equation for all values of $p$

End of section