Difference between revisions of "Scorer Gi"
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| + | The Scorer $\mathrm{Gi}$ function is a solution of the [[differential equation]] $y''(x)-x y(x)=\dfrac{1}{\pi}$ and may be defined by the formula | ||
| + | $$\mathrm{Gi}(x)=\dfrac{1}{\pi} \displaystyle\int_0^{\infty} \sin \left( \dfrac{t^3}{3}+xt \right)dt.$$ | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\mathrm{Gi}(x)=\mathrm{Bi}(x)\displaystyle\int_x^{\infty} \mathrm{Ai}(t)dt + \mathrm{Ai}(x)\displaystyle\int_0^x \mathrm{Bi}(t)dt,$$ | ||
| + | where $\mathrm{Gi}$ denotes the [[Scorer Gi]] function, $\mathrm{Ai}$ denotes the [[Airy Ai]] function, and $\mathrm{Bi}$ denotes the [[Airy Bi]] function. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
=See Also= | =See Also= | ||
[[Airy Ai]]<br /> | [[Airy Ai]]<br /> | ||
[[Airy Bi]]<br /> | [[Airy Bi]]<br /> | ||
[[Scorer Hi]]<br > | [[Scorer Hi]]<br > | ||
Revision as of 17:28, 31 December 2015
The Scorer $\mathrm{Gi}$ function is a solution of the differential equation $y(x)-x y(x)=\dfrac{1}{\pi}$ and may be defined by the formula $$\mathrm{Gi}(x)=\dfrac{1}{\pi} \displaystyle\int_0^{\infty} \sin \left( \dfrac{t^3}{3}+xt \right)dt.$$
Properties
Theorem: The following formula holds: $$\mathrm{Gi}(x)=\mathrm{Bi}(x)\displaystyle\int_x^{\infty} \mathrm{Ai}(t)dt + \mathrm{Ai}(x)\displaystyle\int_0^x \mathrm{Bi}(t)dt,$$ where $\mathrm{Gi}$ denotes the Scorer Gi function, $\mathrm{Ai}$ denotes the Airy Ai function, and $\mathrm{Bi}$ denotes the Airy Bi function.
Proof: █
