Bessel Functions

Definition

Bessel Equation The differential equation below is called the $\nu$ order Bessel equation. $$ \begin{align*} x^2 y^{\prime \prime} +xy^{\prime} +(x^2-\nu^2)y&=0 \\ x(xy^{\prime})^{\prime}+(x^2- \nu ^2) y&=0 \\ y^{\prime \prime}+\frac{1}{x} y^{\prime} + \left( 1-\frac{\nu^{2}}{x^{2}} \right)y&=0 \end{align*} $$

Description

First Kind Bessel Function The first solution of the Bessel equation is written as $J_{\nu}(x)$ and is called the first kind Bessel function. $$ J_{\nu}(x)=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n} }{\Gamma (n+1) \Gamma (n+\nu+1)} \left(\frac{x}{2} \right)^{2n+\nu} $$

$$ J_{-\nu}(x)=\sum \limits_{n=0}^{\infty}\frac{(-1)^{n}}{\Gamma (n+1)\Gamma (n-\nu+1)} \left( \frac{x}{2} \right)^{2n-\nu} $$

Second Kind Bessel Function

Second Kind Bessel Function The second solution of the Bessel equation is referred to as $N_{\nu}(x)=Y_{\nu}(x)$ and is called the second kind Bessel function, Neumann function, or Weber function. 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 $\nu\in \mathbb{Z}$ and $a \in \mathbb{R}\setminus\left\{\mathbb{Z}\right\}$ $$ N_{\nu}(x)=\lim \limits_{a \rightarrow \nu}N_{a}(x) $$

Third Kind Bessel Function

Third Kind Bessel Function The linear combination of the first kind Bessel function and the second kind Bessel function as below is called the third kind Bessel function or Hankel function. $$ H_{p}^{(1)}(x)=J_{p}(x)+iN_{p}(x) \\ H_{p}^{(2)}(x)=J_{p}(x)-iN_{p}(x) $$

Modified Bessel Functions

**Modified Bessel Equation and Modified Bessel Function The differential equation below is called the modified Bessel equation. $$ x^2 y^{\prime \prime} + xy^{\prime}-(x^2-\nu^2)y=0 $$ The solution to the modified Bessel equation is as below and is called the modified Bessel function or the hyperbolic Bessel function.

$$ \begin{align*} I_{\nu}(x)&=i^{-\nu}J_{\nu}(ix) \\ K_{\nu}(x) &= \frac{\pi}{2}i^{\nu+1}\left[ J_{\nu}(ix)+iN_{\nu}(ix) \right] \\ &= \frac{\pi}{2}i^{\nu+1}H_{p}^{(1)}(ix) \\ &=\frac{\pi}{2}\frac{I_{-\nu}(x)-I_{\nu}(x)}{\sin (\nu\pi )} \end{align*} $$

Properties

Symmetry

For integer $\nu$, the following equation holds.

$$ J_{-\nu}(x)=(-1)^{\nu}J_{\nu}(x) $$

Recursion Relationship

Recursion Relation of Bessel Functions $$ \begin{align*} & \frac{d}{dx}[x^{\nu} J_{\nu}(x)] =x^{\nu}J_{\nu-1}(x) \\ & \frac{d}{dx}[x^{-\nu}J_{\nu}(x)]=-x^{-\nu}J_{\nu+1}(x) \\ & J_{\nu-1}(x)+J_{\nu+1}(x)=\frac{2\nu}{x}J_{\nu}(x) \\ & J_{\nu-1}(x)-J_{\nu+1}(x)=2J^{\prime}_{\nu}(x) \\ & J_{\nu}^{\prime}(x)=-\frac{\nu}{x}J_{\nu}(x)+J_{\nu-1}(x)=\frac{\nu}{x}J_{\nu}(x)-J_{\nu+1}(x) \end{align*} $$

Orthogonality

The following is satisfied.

$$ \int_{0}^{1} x J_{\nu}(\alpha x) J_{\nu}(\beta x)dx = \begin{cases} 0 &\alpha\ne \beta \\ \frac{1}{2}J^{2}_{\nu+1}(\alpha)=\frac{1}{2}J_{\nu-1}^{2}(\alpha)=\frac{1}{2}J_{\nu^{\prime}}^{2}(\alpha) &\alpha=\beta \end{cases} $$