# دوال بيسيل

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{{#category_index:B|Bessel functions}} دوال رياضية خاصة تستعمل في المسائل ذات التناظر الاسطواني، خاصة في المعادلات التي تربط عامل لابلاس لدالة بمشتقاتها. انظر Officer (1974, 52–55). سميت نسبة إلى عالم الرياضيات والفلك الألماني فريدريش ويليم بيسيل.

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