1 The Abel operatorThe general Abel integral equation:

has the solution

where is the well-known Abel operator

2 The operatorThe notation is defined as follows:

The operator is defined as:

Obviously, the following two properties for operator are valid

3 The questionProve the following integral equation:

has the following solution

where

and the two-dimensional Laplace operator in polar coordinate, i.e.

4.Hints

To prove the formulae, one may make full use of the two properties of operator and the approach of integration by parts.

These are the hints in the book, but I still can not figure the final formulae out with the help of these hints. So I post it here and wait for your excellent proof.