Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Question

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

$Y$ and $X$ are real variables, $a$ and $b$ are parameters.

Someone said this is a product of polynomial and exponential function.

Do we have a name for this function?

Answer 1

The function $$f: \mathbb R \longrightarrow \mathbb R, f(x):= ax^b\cdot\exp(x)$$ extends to a holomorphic function on the complex plane

$$F: \mathbb C \longrightarrow \mathbb C, F(z):= az^b\cdot\exp(z).$$

Global holomorphic functions are named entire functions. An entire function which is not a polynomial is named a transcendental function. As a consequence, the function $F$ is transcendental. The crucial point is: A polynomial function extends to a holomorphic map $S^2 \longrightarrow S^2$ on the Riemann sphere $S^2$, while a transcendental functions does not.