Definition of incomplete gamma function

Question

I have read following definition of incomplete gamma function $$\Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}dt$$ where $Re(a)>0$. This definition is according to DLMF website (http://dlmf.nist.gov/8.2). I have come across another definition of incomplete Gamma function which is given below $$\int e^{-x^{k}}dx=-\frac{1}{k}\Gamma\left(\frac{1}{k},x^k\right)$$ Now this defintion can be verified from wolfram alpha (https://www.wolframalpha.com/input/?i=int+e%5E(-x%5Ek)). Although I some other knowledgeable person (MSE user with high score) has also used this definition but I have not seen this definition in the literature. I have also tried to find this definition on the internet but I was unsuccessful. I will be very thankful if somebody help me in getting the literature where I can found this second definition. Thanks in advance.

Answer 1

Consider the u-substitution $x=u^{1/k}$.

$$\int e^{-x^k}\ dx=\int e^{-u}\ \frac{du}{ku^{(k-1)/k}}$$

Solving this with the Gamma function gives

$$I=-\frac1k\Gamma\left(\frac1k,u\right)+c=-\frac1k\Gamma\left(\frac1k,x^k\right)+c$$