Dictionary:Bessel functions

{{#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 $m\geq 0$.

External links

find literature about Bessel functions