Difference between revisions of "Scorer Hi"
From specialfunctionswiki
(→Properties) |
|||
| Line 9: | Line 9: | ||
[[Airy Bi]]<br /> | [[Airy Bi]]<br /> | ||
[[Scorer Gi]]<br > | [[Scorer Gi]]<br > | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Revision as of 18:29, 24 May 2016
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: $$\mathrm{Hi}(x)=\mathrm{Bi}(x)\displaystyle\int_{-\infty}^x \mathrm{Ai}(t) \mathrm{d}t - \mathrm{Ai}(x)\displaystyle\int_{-\infty}^x \mathrm{Bi}(t)\mathrm{d}t,$$ where $\mathrm{Hi}$ denotes the Scorer Hi function, $\mathrm{Ai}$ denotes the Airy Ai function, and $\mathrm{Bi}$ denotes the Airy Bi function.
