Difference between revisions of "Airy Bi"
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| − | The Airy function $\mathrm{Bi}$ (sometimes called the "Bairy function") is a solution of the Airy differential equation | + | The Airy function $\mathrm{Bi}$ (sometimes called the "Bairy function") is a solution of the [[Airy differential equation]] |
$$y''(z)-zy(z)=0,$$ | $$y''(z)-zy(z)=0,$$ | ||
which is linearly independent from the [[Airy Ai]] function. | which is linearly independent from the [[Airy Ai]] function. | ||
| Line 10: | Line 10: | ||
</div> | </div> | ||
| + | =Properties= | ||
| + | {{:Relationship between Airy Bi and modified Bessel I}} | ||
=Videos= | =Videos= | ||
Revision as of 05:02, 18 May 2015
The Airy function $\mathrm{Bi}$ (sometimes called the "Bairy function") is a solution of the Airy differential equation $$y(z)-zy(z)=0,$$ which is linearly independent from the Airy Ai function.
- Airybi.png
Aairy $\mathrm{Bi}$ function.
- Complexairybi.png
Domain coloring of analytic continuation of $\mathrm{Bi}$.
Properties
Theorem
The following formula holds: $$\mathrm{Bi}(z)=\sqrt{\dfrac{z}{3}} \left( I_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}} \right) + I_{-\frac{1}{3}} \left( \frac{2}{3} x^{\frac{3}{2}} \right) \right),$$ where $\mathrm{Bi}$ denotes the Airy Bi function and $I_{\nu}$ denotes the modified Bessel $I$.
Proof
References
Videos
Airy differential equation
Series solution of ode: Airy's equation
Leading Tsunami wave reaching the shore
References
The mathematics of rainbows
Tables of Weyl Fractional Integrals for the Airy Function
Special Functions: An Introduction to the Classical Functions of Mathematical Physics
Airy function zeros
