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.