Editing Airy function (section)

==Complex arguments==
We can extend the definition of the Airy function to the complex plane by
<math display="block">\operatorname{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\tfrac{t^3}{3} - zt\right)\, dt,</math>
where the integral is over a path ''C'' starting at the point at infinity with argument {{math|−''π''/3}} and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation {{math|1=''y''′′ − ''xy'' = 0}} to extend {{math|Ai(''x'')}} and {{math|Bi(''x'')}} to [[entire function]]s on the complex plane.

The asymptotic formula for {{math|Ai(''x'')}} is still valid in the complex plane if the principal value of {{math|''x''<sup>2/3</sup>}} is taken and {{mvar|x}} is bounded away from the negative real axis. The formula for {{math|Bi(''x'')}} is valid provided {{mvar|x}} is in the sector <math>x\in\C : \left|\arg(x)\right| < \tfrac{\pi}{3} - \delta</math> for some positive δ. Finally, the formulae for {{math|Ai(−''x'')}} and {{math|Bi(−''x'')}} are valid if {{math|''x''}} is in the sector <math>x\in\C : \left|\arg(x)\right| < \tfrac{2\pi}{3} - \delta.</math>

It follows from the asymptotic behaviour of the Airy functions that both {{math|Ai(''x'')}} and {{math|Bi(''x'')}} have an infinity of zeros on the negative real axis. The function {{math|Ai(''x'')}} has no other zeros in the complex plane, while the function {{math|Bi(''x'')}} also has infinitely many zeros in the sector <math>z\in\C : \tfrac{\pi}{3} < \left|\arg(z)\right| < \tfrac{\pi}{2}.</math>

===Plots===

{| style="text-align:center" align=center
! <math>\Re \left[ \operatorname{Ai} ( x + iy) \right] </math>
! <math>\Im \left[ \operatorname{Ai} ( x + iy) \right] </math>
! <math>\left| \operatorname{Ai} ( x + iy) \right| \, </math>
! <math>\operatorname{arg} \left[ \operatorname{Ai} ( x + iy) \right] \, </math>
|-
|[[File:AiryAi Real Surface.png|200px]]
|[[File:AiryAi Imag Surface.png|200px]]
|[[File:AiryAi Abs Surface.png|200px]]
|[[File:AiryAi Arg Surface.png|200px]]
|-
|[[File:AiryAi Real Contour.svg|200px]]
|[[File:AiryAi Imag Contour.svg|200px]]
|[[File:AiryAi Abs Contour.svg|200px]]
|[[File:AiryAi Arg Contour.svg|200px]]
|}

{| style="text-align:center" align=center
! <math>\Re \left[ \operatorname{Bi} ( x + iy) \right] </math>
! <math>\Im \left[ \operatorname{Bi} ( x + iy) \right] </math>
! <math>\left| \operatorname{Bi} ( x + iy) \right| \, </math>
! <math>\operatorname{arg} \left[ \operatorname{Bi} ( x + iy) \right] \, </math>
|-
|[[File:AiryBi Real Surface.png|200px]]
|[[File:AiryBi Imag Surface.png|200px]]
|[[File:AiryBi Abs Surface.png|200px]]
|[[File:AiryBi Arg Surface.png|200px]]
|-
|[[File:AiryBi Real Contour.svg|200px]]
|[[File:AiryBi Imag Contour.svg|200px]]
|[[File:AiryBi Abs Contour.svg|200px]]
|[[File:AiryBi Arg Contour.svg|200px]]
|}