We investigate the functions $F:\mathbb R\to \mathbb R$ which are $C^\infty $ solutions of Abel functional equation $F(e^x)= F(x)+1$. In particular, we determine asymptotic behaviour derivatives and show that no solution can have $F’$ compl

We suggest an approach for description of integrable cases of the Abel equations using the procedure of increasing the order and equivalence transformations for the induced secondorder equations. A diversity of methods were developed to date for finding solutions of nonlinear ordinary differential equations (ODE). Everybody who encounters integration of a particular ODE uses, as a rule, the acc...

We consider the Abel Equation dy dz = p(z)y2 + q(z)y3 as an equation on the complex plane with p, q – rational functions. The center problem for this equation (which is closely related to the classical center problem for polynomial vector fields on the plane) is to find conditions on p and q under which all the solutions y(z) are univalued functions along the circle |z| = 1. In [3] we have show...

In this paper I present three new results of astronomical interest concerning the theory of Abel inversion. 1) I show that in the case of a spatial emissivity that is constant on toroidal surfaces and projected along the symmetry axis perpendicular to the torus’ equatorial plane, it is possible to invert the projection integral. From the surface (i.e. projected) brightness profile one then form...

In his last letter to Crelle, Abel states a criterion for the solvability by radicals of an irreducible equation prime degree. Sylow finds Abel’s statement ambiguous, and writes that it should be modified. We show correctness original statement.