22 Bessel
Bessel’s DE¶
Family of differential equation, with some constant value \(p\)
\[ x^2y'' + xy' + (x^2-p^2) y = 0 \]
Bessel’s Function¶
is the solution of Bessel’s DE. Denoted by \(J_p(x)\)
\(x=0\) is a regular singular point of equation. Solving using Frobenieus Series method gives 2 initial roots as \(m = \pm p\)
| \(+p\) | \(-p\) | |
|---|---|---|
| \(J(x)\) | \(\sum\limits_{n=0}^\infty \dfrac{(-1)^n \left(\frac{x}{2}\right)^{2n \textcolor{hotpink}{+p}}}{n!(n \textcolor{hotpink}{+p})!}\) | \(\sum\limits_{n=0}^\infty \dfrac{(-1)^n \left(\frac{x}{2}\right)^{2n \textcolor{hotpink}{-p} }}{n!(n \textcolor{hotpink}{-p} )!}\) |
The above 2 formula are not directly possible for negative integers, as \((n-p)!\) is not valid when it is negative
Use gamma function
General Solution¶
\[ y = c_1 J_p(x) + c_2 J_{-p} (x) \]
Properties¶
To Remember¶
\[ \begin{aligned} J_\frac{1}{2}(x) &= \sin x \sqrt{ \frac{2}{\pi x} } \\ J_\frac{-1}{2}(x) &= \cos x \sqrt{ \frac{2}{\pi x} } \\ J_{p-1}(x) + J_{p+1}(x) &= \frac{2p}{x} J_p(x) \end{aligned} \]
Other Properties¶
\[ \begin{aligned} \Big( x^{p} J_p(x) \Big)' &= x^{p} J_{p-1} (x) \\ \Big( x^{-p} J_p(x) \Big)' &= - x^{-p} J_{p+1} (x) \\ {J_p}'(x) + \frac{p}{x} J_p(x) &= J_{p-1}(x) \\ {J_p}'(x) - \frac{p}{x} J_p(x) &= - J_{p+1}(x) \\ J_{p-1}(x) - J_{p+1}(x) &= 2 {J_p}'(x) \end{aligned} \]