Nontangential limit

Remarks

The limit along $\Gamma_\alpha(\zeta)$ is a limit of a function of several variables restricted to that set; $\zeta$ is a limit point of each $\Gamma_\alpha(\zeta)$. For the unit disc the regions $\Gamma_\alpha(\zeta)$ are the Stolz angles, and the same definition works for the upper half-space with cones $\set{(x, t) : \abs{x - x_0} < \alpha t}$. A nontangential limit implies the radial limit $\lim_{r \to 1} u(r\zeta) = L$. Example: let $f(z) = \exp\left(-\frac{1 + z}{1 - z}\right)$ on the unit disc and $\zeta = 1$. Since $\operatorname{Re} \frac{1 + z}{1 - z} = \frac{1 - \abs{z}^2}{\abs{1 - z}^2}$, on $\Gamma_\alpha(1)$ this is at least $\frac{1}{\alpha^2 (1 - \abs{z})}$, which tends to $+\infty$. So $f$ has nontangential limit $0$ at $1$. On the circle $\abs{z - 1/2} = 1/2$ the same real part equals $1$, so $\abs{f} = e^{-1}$ there, and $f$ has no unrestricted limit at $1$. Non-example: the Poisson kernel $u(z) = \frac{1 - \abs{z}^2}{\abs{1 - z}^2}$ has $u(r) = \frac{1 + r}{1 - r} \to +\infty$, so it has no finite nontangential limit at $1$. By Fatou's theorem, the Poisson integral of an integrable function has a nontangential limit at almost every boundary point.