Revision as of 07:45, 16 May 2016

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.

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

Theorem

The following formula holds: $$\mathrm{Gi}(x)=\mathrm{Bi}(x)\displaystyle\int_x^{\infty} \mathrm{Ai}(t)\mathrm{d}t + \mathrm{Ai}(x)\displaystyle\int_0^x \mathrm{Bi}(t) \mathrm{d}t,$$ where $\mathrm{Gi}$ denotes the Scorer Gi function, $\mathrm{Ai}$ denotes the Airy Ai function, and $\mathrm{Bi}$ denotes the Airy Bi function.

Proof

References

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.

Proof

References

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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