Scorer Hi
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The Scorer $\mathrm{Hi}$ function is a solution of the differential equation $y(x)-x y(x)=\dfrac{1}{\pi}$ and may be defined by the formula $$\mathrm{Hi}(x)=\dfrac{1}{\pi} \displaystyle\int_0^{\infty} \exp \left( -\dfrac{t^3}{3}+xt \right)dt.$$
Properties
Theorem: The following formula holds: $$\mathrm{Gi}(x)=\mathrm{Bi}(x)\displaystyle\int_{-\infty}^x \mathrm{Ai}(t)dt - \mathrm{Ai}(x)\displaystyle\int_{-\infty}^x \mathrm{Bi}(t)dt,$$ where $\mathrm{Hi}$ denotes the Scorer Gi function, $\mathrm{Ai}$ denotes the Airy Ai function, and $\mathrm{Bi}$ denotes the Airy Bi function.
Proof: █
