Bessel Differential Equation The Bessel differential equation is the linear second-order ordinary differential equation given by ( 1 ) Equivalently, dividing through by , ( 2 ) The solutions to this equation define the Bessel functions and . The equation has a regular singularity at 0 and an irregular singularity at . A transformed version of the Bessel differential equation given by Bowman (1958) is ( 3 ) The solution is ( 4 ) where ( 5 )
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Bessel Differential Equation
The Bessel differential equation is the linear second-order ordinary differential equation given by
(1)
Equivalently, dividing through by ,
(2)
The solutions to this equation define the Bessel functions and . The equation has a regular singularity at 0 and an irregular singularity at .
A transformed version of the Bessel differential equation given by Bowman (1958) is
(3)
The solution is
(4)
where
(5)
and are the Bessel functions of the first and second kinds, and and are constants. Another form is given by letting , , and (Bowman 1958, p. 117), then
The Bessel functions are more frequently defined as solutions to the differential equation
(3)
There are two classes of solution, called the Bessel function of the first kind and Bessel function of the second kind . (A Bessel function of the third kind, more commonly called a Hankel function, is a special combination of the first and second kinds.) Several related functions are also defined by slightly modifying the defining equations.
The Bessel functions of the first kind are defined as the solutions to the Bessel differential equation
(1)
which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows for , 1, 2, ..., 5. The notation was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function
(2)
is the same as the modern one (Watson 1966, p. 14). Bessel used the notation to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279).
The Bessel function can also be defined by the contour integral
(3)
where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
The Bessel function of the first kind is implemented in Mathematica as BesselJ[nu, z].
But for , so the denominator is infinite and the terms on the left are zero. We therefore have
(50)
(51)
Note that the Bessel differential equation is second-order, so there must be two linearly independent solutions. We have found both only for . For a general nonintegral order, the independent solutions are and . When is an integer, the general (real) solution is of the form
(52)
where is a Bessel function of the first kind, (a.k.a. ) is the Bessel function of the second kind (a.k.a. Neumann function or Weber function), and and are constants. Complex solutions are given by the Hankel functions (a.k.a. Bessel functions of the third kind).
The Bessel functions are orthogonal in according to
(53)
where is the th zero of and is the Kronecker delta (Arfken 1985, p. 592).
(Abramowitz and Stegun 1972, p. 360), or the integral
(79)
Bessel Function of the Second Kind
A Bessel function of the second kind (e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted (e.g, Gradshteyn and Ryzhik 2000, p. 657, eqn. 6.518), is a solution to the Bessel differential equation which is singular at the origin. Bessel functions of the second kind are also called Neumann functions or Weber
functions. The above plot shows for , 1, 2, ..., 5. The Bessel function of the second kind is implemented in Mathematica as BesselY[nu, z].
Let be the first solution and be the other one (since the Bessel differential equation is second-order, there are two linearly independent solutions). Then
(1) (2)
Take (1) minus (2),
(3) (4)
so , where is a constant. Divide by ,
(5) (6)
Rearranging and using gives
(7) (8)
where is the so-called Bessel function of the second kind.