Difference between revisions of "Airy Bi"
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The Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ solve the differential equation | The Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ solve the differential equation | ||
$$y'' - xy = 0.$$ | $$y'' - xy = 0.$$ | ||
| + | |||
| + | It can be shown that | ||
| + | $$\mathrm{Ai}(x) = \dfrac{1}{\pi} \displaystyle\int_0^{\infty} \cos \left( \dfrac{t^3}{3} + xt \right) dt$$ | ||
| + | and | ||
| + | $$\mathrm{Bi}(x) = \dfrac{1}{\pi} \displaystyle\int_0^{\infty} \left[ e^{-\frac{t^3}{3} + xt} + \sin \left( \dfrac{t^3}{3}+xt \right) \right] dt.$$ | ||
Revision as of 01:00, 3 August 2014
The Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ solve the differential equation $$y - xy = 0.$$
It can be shown that $$\mathrm{Ai}(x) = \dfrac{1}{\pi} \displaystyle\int_0^{\infty} \cos \left( \dfrac{t^3}{3} + xt \right) dt$$ and $$\mathrm{Bi}(x) = \dfrac{1}{\pi} \displaystyle\int_0^{\infty} \left[ e^{-\frac{t^3}{3} + xt} + \sin \left( \dfrac{t^3}{3}+xt \right) \right] dt.$$
