I'm using the integral calculator to calculate $\int_{-\infty}^{+\infty}e^{-x^4}dx$. The antiderivative that the calculator shows is this, a gamma function with two input values:
$$ - \frac{\Gamma\left(\frac 14, x^4\right)x}{4|x|} +C $$
I don't know much about the gamma function. I tried reading its Wikipedia and I could not find any definition of the gamma function with two input values. So I'm here to ask if you can give me a reference to read about.
To see the connection to the Gamma function, it's better to look at the integral $\int_0^\infty e^{-x^4}\,dx$. (This is half the original integral by symmetry, so this isn't much of a change.) Then the substitution $u=x^4$ converts this integral to $\frac14 \int_0^\infty u^{-3/4} e^{-u}du$. But the Gamma function is defined as $$\Gamma(z)=\int_0^\infty e^{-u} u^{z-1}\,du$$ so the integral of interest is $\frac14 \Gamma(1/4)$. Hence the original integral is $\frac12 \Gamma(1/4)$. This approach works more generally for integrands of the form $e^{-x^k}$, though in that case one should restrict to $[0,\infty)$ alone.