Solving an integral equation

**1 The Abel operator**

The general Abel integral equation:

has the solution

where is the well-known Abel operator

**2 The operator**

The notation is defined as follows:

The operator is defined as:

Obviously, the following two properties for operator are valid

**3 The question**

Prove 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.