Zero and first order Hankel Transforms

Zero-order_Hankel_transform_and_its_inverseIn summary, the conversation discusses a problem involving the first and zero order Hankel transforms in cylindrical symmetry. The question is if there is a way to analytically take the inverse Hankel transform of the zero order Hankel transform of a function. The solution can be found by plugging in the definitions of the transforms.
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dnp
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Hello everyone,

I am dealing with the following problem. Solving and kinetic equation I came up with the expression

H_1^(-1)[H_0(P(r))/q]

where H_0 is the zero order Hankel transform, H_1^(-1) is the first order inverse Hankel transform P(r) is a function that depends on the radial coordinate in cylindrical symmetry. q is the variable of the transformation that comes from the definitions of the respective Bessel functions J_0(q*r) and J_1(q*r).

My question is is there a way to analytically take an inverse first order Hankel transform of a zero order Hankle transform of the function P(r). I have been through a lot of literature but could not find any discussion. On the other hand such expression seems to occeru pretty commonly when dealing with the radial part of cylindrical divergence.

Thanks in advance for any help

dnp
 
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FAQ: Zero and first order Hankel Transforms

What is a Hankel Transform?

A Hankel Transform is a mathematical operation that converts a function in one domain (usually in terms of radial distance) into a function in another domain (usually in terms of frequency or wavenumber). It is commonly used in signal processing and image reconstruction.

What is the difference between zero and first order Hankel Transforms?

The difference lies in the order of the Bessel function used in the transformation. In zero order Hankel Transform, the Bessel function of the first kind is used, while in the first order Hankel Transform, the Bessel function of the second kind is used. This leads to different mathematical properties and applications of the two transforms.

What are some common applications of Hankel Transforms?

Hankel Transforms are commonly used in solving differential equations, image reconstruction in tomography, and in signal processing. They are also used in optics and acoustics for analyzing wave propagation.

How is the inverse Hankel Transform obtained?

The inverse Hankel Transform is obtained by using the inverse Bessel function of the corresponding order. In zero order Hankel Transform, the inverse Bessel function of the first kind is used, while in the first order Hankel Transform, the inverse Bessel function of the second kind is used.

Can Hankel Transforms be applied to functions in higher dimensions?

Yes, Hankel Transforms can be extended to functions in higher dimensions, such as in cylindrical and spherical coordinates. In these cases, the order of the Bessel function used may change, but the general principles of the transform remain the same.

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