Tricomi confluent hypergeometric function

Remarks

The function $U(a, b, \cdot)$ solves Kummer's equation, since both terms of the definition do. It is characterised among the solutions by $U(a, b, z) \sim z^{-a}$ as $z \to \infty$ with $\abs{\arg z} < \frac{3\pi}{2} - \delta$, for each $\delta > 0$. For $\operatorname{Re} a > 0$ and $\operatorname{Re} z > 0$ it has the integral form $$U(a, b, z) = \frac{1}{\Gamma(a)} \int_0^\infty e^{-zt}\, t^{a - 1} (1 + t)^{b - a - 1}\, dt.$$ Example: $U(a, a + 1, z) = z^{-a}$; for $\operatorname{Re} a > 0$ and $\operatorname{Re} z > 0$ the integral is $\frac{1}{\Gamma(a)} \int_0^\infty e^{-zt} t^{a - 1}\, dt = z^{-a}$. Non-example: $M(a, b, z)$ itself is in general not a multiple of $U(a, b, z)$; for $a = b$ it is $e^z$, which grows faster than any power as $z \to +\infty$. The two functions satisfy Kummer's second transformation $U(a, b, z) = z^{1 - b} U(a - b + 1, 2 - b, z)$.