ABSTRACT
Throughout this thesis, we use some nonstandard concepts to study
the analyticity near the singularity. We analyzed and proved the
existence and uniqueness theorems for first order ordinary differential
equations in subset of the monad of the initial standard point.
Then the solutions of the second order ordinary differential equation
(Legendre Equation) are introduced around the singularity in the monad
of zero by using power series method with suitable transformations for
singular points. Additionally, we wrote the Legendre polynomial on
the form of Mehler-Dirichlet Integral formula to find its solutions near
the singularity.
Furthermore, nonstandard analysis tools are successfully applied
to find a nonstandard analytic solution for the first order differential
equation near singularity.
Finally, we studied in details the proof of Painlev´e’s Theorem in
nonstandard analysis, and we get more precise result than that exists
in the conventional (classical) version.

 

 

 

 

24/11/2016