Difference between revisions of "Airy Bi"

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The Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ solve the differential equation
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__NOTOC__
$$y'' - xy = 0.$$
 
  
It can be shown that
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The Airy function $\mathrm{Bi}$ (sometimes called the "Bairy function") is a solution of the [[Airy differential equation]]
$$\mathrm{Ai}(x) = \dfrac{1}{\pi} \displaystyle\int_0^{\infty} \cos \left( \dfrac{t^3}{3} + xt \right) dt$$
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$$y' '(z)-zy(z)=0,$$
and
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which is [[linearly independent]] from the [[Airy Ai]] function.
$$\mathrm{Bi}(x) = \dfrac{1}{\pi} \displaystyle\int_0^{\infty} \left[ e^{-\frac{t^3}{3} + xt} + \sin \left( \dfrac{t^3}{3}+xt \right) \right] dt.$$
 
  
[[File:Airyai.png|500px]]
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<div align="center">
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<gallery>
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File:Airybiplot.png|Aairy $\mathrm{Bi}$ function.
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File:Complexairybiplot.png|[[Domain coloring]] of $\mathrm{Bi}$.
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</gallery>
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</div>
  
[[File:Airybi.png|500px]]
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=Properties=
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[[Relationship between Airy Bi and modified Bessel I]]<br />
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[[Relationship between Scorer Gi and Airy functions]]<br />
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[[Relationship between Scorer Hi and Airy functions]]<br />
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=Videos=
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[https://www.youtube.com/watch?v=HlX62TkR6gc&noredirect=1 Leading Tsunami wave reaching the shore (27 November 2009)]<br />
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[https://www.youtube.com/watch?v=0jnXdXfIbKk&noredirect=1 Series solution of ode: Airy's equation (3 November 2010)]<br />
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[https://www.youtube.com/watch?v=oYJq3mhg5yE&noredirect=1 Airy differential equation (26 November 2013)]<br />
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=References=
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[http://www.ams.org/samplings/feature-column/fcarc-rainbows The mathematics of rainbows]<br />
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[http://www.ams.org/journals/mcom/1979-33-145/S0025-5718-1979-0514831-8/S0025-5718-1979-0514831-8.pdf Tables of Weyl Fractional Integrals for the Airy Function]<br />
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[http://www.amazon.com/Special-Functions-Introduction-Classical-Mathematical/dp/0471113131 Special Functions: An Introduction to the Classical Functions of Mathematical Physics]<br />
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[http://www.people.fas.harvard.edu/~sfinch/csolve/ai.pdf Airy function zeros]
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=See Also=
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[[Airy Ai]] <br />
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[[Scorer Gi]] <br />
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[[Scorer Hi]] <br />
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[[Category:SpecialFunction]]

Latest revision as of 16:07, 21 October 2017


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

Relationship between Airy Bi and modified Bessel I
Relationship between Scorer Gi and Airy functions
Relationship between Scorer Hi and Airy functions

Videos

Leading Tsunami wave reaching the shore (27 November 2009)
Series solution of ode: Airy's equation (3 November 2010)
Airy differential equation (26 November 2013)

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

See Also

Airy Ai
Scorer Gi
Scorer Hi