Proximate type in reference to generalized biaxisymmetric potentials (Q2770150)

The analysis of functions \(F^{(\alpha,\beta)}(x,y)\), solutions of the equation NEWLINE\[NEWLINE\frac{\partial^2F}{\partial x^2}+ \frac{\partial^2F}{\partial y^2}+ \frac{2\alpha+1}{x} \frac{\partial F}{\partial x}+ \frac{2\beta+1}{y} \frac{\partial F}{\partial y}=0, \qquad \alpha> \beta> -1/2,NEWLINE\]NEWLINE in the Cauchy data \(F_x^{(\alpha,\beta)} (0,y)= F_y^{(\alpha,\beta)} (x,0)= 0\) along the singular lines in the hypersphere \(\Sigma_R^{(\alpha,\beta)}: x^2+ y^2< R^2\) is the object of the paper. The author sees in the above functions, called by him generalized biaxially symmetric potentials, ``natural extensions of harmonic or analytic functions'' and the study is a continuation of other ones, published previously by him and by another author. In the present paper he introduces a proximate type function \(T(r)\), \(0< r< R\), \(0< R< \infty\) by means of which he analyzes the behaviour and the growth properties of \(F^{(\alpha,\beta)}\), in the third part of the paper. Previously, in the second part of it, he establishes some properties of \(T(r)\), useful for the demonstration of the \(F^{(\alpha,\beta)}\) properties.