Bessel Functions
#Math
The orthogonal basis set for cylindrical coordinates
$\displaystyle j_{l}(x)\equiv (-x)^{l}\left( \frac{1}{x}\frac{\mathrm{d} }{ \mathrm{d}x} \right)^{l} \frac{\sin x}{x}\rightarrow \frac{2^{l}l!}{(2l+1)!}x^{l}\text{ for }x\ll 1$
- Used to solve the infinite sphere well
Example Values
| Bessel Function | Value |
|---|---|
| $\displaystyle j_{0}(x)$ | $\displaystyle \frac{\sin x}{x}$ |
| $\displaystyle j_{1}(x)$ | $\displaystyle \frac{\sin x}{x^{2}}-\frac{\cos x}{x}$ |
| $\displaystyle j_{2}(x)$ | $\displaystyle \left(\frac{3}{x^{3}}-\frac{1}{x}\right)-\frac{3}{x^{2}}\cos x$ |