# Dictionary:Bessel functions

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(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

${\displaystyle z^{2}y''+zy'+(z^{2}-\alpha ^{2})y=0.}$

Here ${\displaystyle \alpha }$ need not be an integer and ${\displaystyle y=y(z)}$

Another form of this equation may be obtained by dividing through by the coefficient ${\displaystyle z^{2}}$

${\displaystyle y''+{\frac {1}{z}}y'+\left(1-{\frac {\alpha ^{2}}{z^{2}}}\right)y=0.}$

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

${\displaystyle y=z^{\alpha }\sum _{n=0}^{\infty }a_{n}z^{n}=\sum _{n=0}^{\infty }a_{n}z^{n+\alpha }}$

is substituted into the Bessel differential equation and the coefficients ${\displaystyle a_{n}}$ are found be equating terms of like powers in ${\displaystyle z^{m}.}$

The result of this procedure yields, in general, the series solution

${\displaystyle J_{\alpha }(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(z/2)^{2n+\alpha }}{n!\;\Gamma (n+\alpha +1)}}.}$

Here ${\displaystyle \Gamma (n+\alpha +1)}$ is the Gamma Function being used as an analytic continuation of the factorial function.

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

${\displaystyle J_{m}(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(z/2)^{2n+m}}{n!\;[n+m]!}}}$

and this also holds for negative values of ${\displaystyle m}$

${\displaystyle J_{-m}(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(z/2)^{2n+m}}{n!\;[n+m]!}}.}$

Here ${\displaystyle m\geq 0}$.