Airy Bi

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 $I$.

Videos

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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