Two definitions of the Incomplete Gamma Function - are they equivalent?

Question

From Loss Models, 4th ed., by Klugman et al.:

Definition 5.5 The incomplete Gamma function with parameter $\alpha > 0$ is denoted and defined by $$\Gamma\left(\alpha ; x\right) = \dfrac{1}{\Gamma(\alpha)}\int\limits_{0}^{x}t^{\alpha - 1}e^{-t}\text{ d}t\text{.}$$

I assume this is the "upper" incomplete Gamma function I've read about online. From Wolfram MathWorld:

$$\Gamma(\alpha;x) = \int\limits_{x}^{\infty}t^{a-1}e^{-t}\text{ d}t\text{.}$$

Are these definitions equivalent?

Answer 1

No, they're not equivalent.

The function in Klugman et al. should perhaps be called the normalized lower incomplete gamma function, and the function on MathWorld is commonly called the upper incomplete gamma function.

See Wikipedia and the DLMF.