How to evaluate Euler-type integral

Question

Suppose $f(x)=int_{0}^x e^{-t}t^{-1/2}dt$. If $x to infty$, this is the gamma function. But how to compute this integral if $t$ is not $infty$? Is there any convenient implementation in software such as R?

Claude Leibovici · Answer

Let $t=u^2$
$$I=int_0^xfrac{e^{-t}}{sqrt{t}},dt=2int_0^{sqrt x} e^{-u^2},du=sqrt{pi } ,text{erf}left(sqrt{x}right)$$
This error function does exist in $ R$

How to evaluate Euler-type integral

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