Axially symmetric multi-instanton solutions generated by conformal mappings

It is shown that exact axially symmetric, regular self-dual multi-instanton solutions of the SU(2) Yang-Mills equations can be generated from the vacuum solution by means of singular conformal mappings of a complex plane followed by a singular gauge transformation. The regularizing gauge for the 't Hooft singular solution is found and the corresponding Yang-Mills gauge potential ${\mathit{A}}_{\mu }$ without any singularity and with the correct asymptotic behavior is given. The relations between the parameters in different forms of the solution are explored and the possible implications on the calculations of quantum effects are noted.