Abel function
Encyclopedia
In mathematics Abel function is a special kind of solution of the Abel equations, used to classify them as superfunction
s, and formulate conditions of uniqueness.
The Abel equation
is class of equations which can be written in the form
where function ƒ is supposed to be given, and function g is expected to be found. This equation is closely related to the iterative equation
which is also called "Abel equation".
In general the Abel equation may have many solutions, and the additional requirements are necessary to select the only one among them.
, 
is holomorphic function
on
, 
is holomorphic function
on
Then and only then
is
superfunction
of
on 
is 
superfunction on 
on 
, 
is holomorphic on 
Then and only then
id 
Abel function in 
with respect to 
on 
.
Superfunction
In mathematics, a superfunction is a nonstandard name for an iterated function for complexified continuous iteration index.Roughly, for some function f and for some variable x, the superfunction could be defined by the expression S =...
s, and formulate conditions of uniqueness.
The Abel equation
Abel equation
The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the formf = h\,\!or\alpha=\alpha+1\!and shows non-trivial properties at the iteration.-Equivalence:...
is class of equations which can be written in the form
where function ƒ is supposed to be given, and function g is expected to be found. This equation is closely related to the iterative equation
which is also called "Abel equation".
In general the Abel equation may have many solutions, and the additional requirements are necessary to select the only one among them.
Definition 1: Superfunction
If, is holomorphic functionHolomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
on
, is holomorphic functionHolomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
on
Then and only then
is superfunctionSuperfunction
In mathematics, a superfunction is a nonstandard name for an iterated function for complexified continuous iteration index.Roughly, for some function f and for some variable x, the superfunction could be defined by the expression S =...
of
on Definition 2: Abel function
If is superfunction on on , is holomorphic on Then and only then
id Abel function in with respect to on .