Bessel Function and Hankel Transform

2. The Bessel function are a series of
solutions to a second order differential
equation 𝑥2
𝑦′′
+ 𝑥𝑦′
+ 𝑥2
+ 𝑝2
𝑦 = 0 that
arises in many diverse situation.

3.  The Hankel Transform is an integral
transform and was developed by the
mathematician Herman Hankel which occurs
in the study of function which depend only on
the distance from the origin. It is also known
as the Fourier-Bessel Transform.

4.  Hankel transform expresses any given
function f(r) as the weighted sum of an
infinite number of Bessel functions of the
first kind Jν(kr).

5.  The Hankel transform appears when one
write the multidimensional Fourier transform
in hypo spherical coordinate, which is the
reason why the Hankel transform often
appears in physical problems with cylindrical
or spherical symmetry.

6. Recall the Bessel equation
𝑥2 𝑦′′ + 𝑥𝑦′ + 𝑥2 + 𝑛2 𝑦 = 0
For a fixed value of n, this equation has two linearly
independent solutions. One of these solutions, that can be
obtained using Frobenius' method, is called a Bessel
function of the first kind, and is denoted by Jn(x). This
solution is regular at x = 0. The second solution, that is
singular at x = 0,is called a Bessel function of the second
kind, and is denoted by Yn(x).

7. The use of Hankel transform has many
advantages.
 It is applicable to both homogeneous and
inhomogeneous problems.
 It simplify calculations and singles out the
purely computational part of solution.s