# $\int_a^\infty\frac{x^t}{\Gamma(t+1)}dt$

Let $$I(x,a)=\int_a^\infty\frac{x^t}{\Gamma(t+1)}dt$$. $$I(x,a)$$ converges for $$x\in\mathbb{R}^+, a\in\mathbb{R}$$. Note $$\frac{\partial}{\partial x}I(x,a)=I(x,a-1)=I(x,a)+\int_{a-1}^a\frac{x^t}{\Gamma(t+1)}dt$$ This integral was inspired by generalizing the Taylor series of $$\exp(x)$$ to a more continuous case.

I sincerely hope that someone has thought of this integral in more detail and will share further insight.