# Bessel functions  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro] Explicit plots of the $J_{0}$ and $J_{1}$ from Complex map of $J_{1}$ by ; $u+\mathrm {i} v=J_{1}(x+\mathrm {i} y)$ .

Bessel functions are solutions of the Bessel differential equation:

$z^{2}{\frac {d^{2}w}{dz^{2}}}+z{\frac {dw}{dz}}+(z^{2}-\alpha ^{2})w=0$ where α is a constant.

Because this is a second-order differential equation, it should have two linearly-independent solutions:

(i) Jα(x) and
(ii) Yα(x).

In addition, a linear combination of these solutions is also a solution:

(iii) Hα(x) = C1 Jα(x) + C2 Yα(x)

where C1 and C2 are constants.

These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind.

## Properties

Many properties of functions $J$, $Y$ and $H$ are collected at the handbook by Abramowitz, Stegun .

### Integral representations

$\!\!\!\!\!\!\!\!\!\!(9.1.20)~~~\displaystyle J_{\nu }(z)={\frac {(z/2)^{\nu }}{\pi ^{1/2}~(\nu -1/2)!}}~\int _{0}^{\pi }~\cos(z\cos(t))\sin(t)^{2\nu }~t~\mathrm {d} t$ ### Expansions at small argument

$\displaystyle J_{\alpha }(z)=\left({\frac {z}{2}}\right)^{\!\alpha }~\sum _{k=0}^{\infty }~{\frac {(-z^{2}/4)^{k}}{k!~(\alpha \!+\!k)!}}$ The series converges in the whole complex $z$ plane, but fails at negative integer values of $\alpha$ . The postfix form of factorial is used above; $k!=\mathrm {Factorial} (k)$ .

## Applications

Bessel functions arise in many applications. For example, Kepler’s Equation of Elliptical Motion, the vibrations of a membrane, and heat conduction, to name a few. In paraxial optics the Bessel functions are used to describe solutions with circular symmetry.