# Dictionary:Bessel functions

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{{#category_index:B|Bessel functions}} (bes’ ∂l) Special mathematical functions that often occur in problems involving cylindrical symmetry, especially in equations relating the Laplacian of a function to derivatives of the function. See Officer (1974, 52–55). Named for Friedrich Wilhelm Bessel (1784–1846), German astronomer and mathematician.

## The Bessel ordinary differential equation

Bessel functions are particular solutions to the Bessel ordinary differential equation

$z^{2}y''+zy'+(z^{2}-\alpha ^{2})y=0.$ Here $\alpha$ need not be an integer and $y=y(z)$ Another form of this equation may be obtained by dividing through by the coefficient $z^{2}$ $y''+{\frac {1}{z}}y'+\left(1-{\frac {\alpha ^{2}}{z^{2}}}\right)y=0.$ One method of solution of this equation is to apply the Method of Frobenius, wherein a trial solution in the form of an infinite series

$y=z^{\alpha }\sum _{n=0}^{\infty }a_{n}z^{n}=\sum _{n=0}^{\infty }a_{n}z^{n+\alpha }$ is substituted into the Bessel differential equation and the coefficients $a_{n}$ are found be equating terms of like powers in $z^{m}.$ The result of this procedure yields, in general, the series solution

$J_{\alpha }(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(z/2)^{2n+\alpha }}{n!\;\Gamma (n+\alpha +1)}}.$ Here $\Gamma (n+\alpha +1)$ is the Gamma Function being used as an analytic continuation of the factorial function.

For $\alpha =m$ where $m$ is an integer, this reduces to

$J_{m}(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(z/2)^{2n+m}}{n!\;[n+m]!}}$ and this also holds for negative values of $m$ $J_{-m}(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(z/2)^{2n+m}}{n!\;[n+m]!}}.$ Here $m\geq 0$ .