蔷薇少女馆童真誉失忆:Bessel's Differential Equation
In the Sturm-Liouville Boundary Value Problem, there is an important special case called Bessel's Differential Equation which arises in numerous problems, especially in polar and cylindrical coordinates. Bessel's Differential Equation is defined as:
where
Since Bessel's differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind
However,
See plots of Bessel Functions
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Important PropertiesBasic Relationship: The Bessel function of the first kind of order
The Bessel function of the second kind of order
Generating Function: The generating function of the Bessel Function of the first kind is
Recurrence Relation: A Bessel function of higher order can be expressed by Bessel functions of lower orders.
Asymptotic Approximations: Keeping the first few terms in the series expansions, the behavior of a Bessel function at small or large
For small
For large
associated with a physical problem defined on the interval of
By using this orthogonality, the
The general solution thus yields
where
and
This orthogonal series expansion is also known as a Fourier-Bessel Series expansion or a Generalized Fourier Series expansion. The transform based on this relationship is called a Hankel Transform.
Hankel Function: Similar to
For large
Modified Bessel Function: Similar to the relations between the trigonometric functions and the hyperbolic trigonometric functions,
The modified Bessel functions of the first and second kind of order
See further detail on the modified Bessel functions.
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Special Results