Airy Functions for Complex Numbers

Calculation of Ai(z) and Bi(z) - Solutions of the Airy Differential Equation

Airy Functions Calculator

Airy Functions

The Airy functions Ai(z) and Bi(z) are two linearly independent solutions of the Airy differential equation \(y'' - zy = 0\). They play an important role in optics, quantum mechanics, and electromagnetism.

Argument z = a + bi
+
i
Calculation Results
Ai(z) =
Bi(z) =

Airy Functions - Properties

Differential Equation
\[y'' - zy = 0\]

Airy equation (Stokes equation)

Two Solutions
Ai(z) and Bi(z)
are linearly independent
Ai(z) 1st Kind
Bi(z) 2nd Kind
Important Properties
  • Ai(z) → 0 for z → +∞ (decaying)
  • Bi(z) → ∞ for z → +∞ (growing)
  • Both oscillate for z < 0
  • Wronskian: \(W = Ai(z)Bi'(z) - Ai'(z)Bi(z) = \frac{1}{\pi}\)
Related Functions

Real Numbers:
Airy functions for real numbers →

Applications
  • Optics: Light diffraction
  • Quantum mechanics: WKB approximation
  • Electromagnetism: Wave propagation
  • Astronomy: Caustics


Formulas for the Airy Functions

The Airy functions can be expressed in terms of modified Bessel functions.

Ai(z) - First Kind
\[Ai(z) = \frac{1}{\pi}\sqrt{\frac{z}{3}}K_{\frac{1}{3}}\left(\frac{2}{3}z^{\frac{3}{2}}\right)\]

With modified Bessel function \(K_{1/3}\)

Bi(z) - Second Kind
\[Bi(z) = \sqrt{\frac{z}{3}}\left(I_{-\frac{1}{3}}\left(\frac{2}{3}z^{\frac{3}{2}}\right) + I_{\frac{1}{3}}\left(\frac{2}{3}z^{\frac{3}{2}}\right)\right)\]

With modified Bessel functions \(I_{\pm 1/3}\)

Airy Functions - Detailed Description

Definition

The Airy functions are named after the British astronomer George Biddell Airy (1801-1892), who used them in his work on optics.

Airy Differential Equation:
\[y'' - zy = 0\]
also called the Stokes equation.

General Solution:
\[y(z) = c_1 Ai(z) + c_2 Bi(z)\]
with arbitrary constants \(c_1, c_2\)

Ai(z) - First Kind

The Airy function of the first kind Ai(z):

Behavior:

  • Ai(z) → 0 for z → +∞ (exponential decay)
  • Ai(z) oscillates for z < 0
  • Ai(0) ≈ 0.35502805...
  • Ai'(0) ≈ -0.25881940...

Bi(z) - Second Kind

The Airy function of the second kind Bi(z):

Behavior:

  • Bi(z) → ∞ for z → +∞ (exponential growth)
  • Bi(z) oscillates for z < 0
  • Bi(0) ≈ 0.61492662...
  • Bi'(0) ≈ 0.44828835...

Wronskian Determinant

Wronskian:
\[W = Ai(z)Bi'(z) - Ai'(z)Bi(z) = \frac{1}{\pi}\]
is constant (independent of z)!

Relations

Symmetry:
Ai(-z) and Bi(-z) are complex functions

Derivatives:
Ai'(z) and Bi'(z) also satisfy the Airy equation

Physical Applications

Optics:
• Light diffraction: Edge diffraction
• Caustics: Focal lines and surfaces
• Rainbow: Intensity distribution
• Airy disk: Diffraction pattern
Quantum Mechanics:
• WKB approximation: Tunneling effect
• Potential well: Linear potential
• Energy levels: Zeros of Ai
• Scattering theory: Asymptotic behavior

Further Applications

Electromagnetism

• Wave propagation in media
• Electromagnetic fields
• Antenna theory

Astronomy

• Caustics in gravitational lenses
• Light propagation
• Stellar atmospheres

Mathematics

• Special functions
• Asymptotic expansion
• Integral equations

Historical Significance

George Biddell Airy introduced these functions in 1838 to describe the intensity distribution when light passes through a circular aperture (Airy disk). The functions later proved to be fundamental in many other areas of physics and mathematics.

Integral Representations

Ai(z) - Integral Form
\[Ai(z) = \frac{1}{\pi}\int_0^\infty \cos\left(\frac{t^3}{3} + zt\right)dt\]

Converges for all z

Contour Integral
\[Ai(z) = \frac{1}{2\pi i}\int_C e^{t^3/3 - zt}dt\]

With suitable integration path C

Asymptotic Expansions

For z → +∞:

\[Ai(z) \sim \frac{1}{2\sqrt{\pi}z^{1/4}}e^{-\frac{2}{3}z^{3/2}}\]

Exponential decay

For z → -∞:

\[Ai(z) \sim \frac{1}{\sqrt{\pi}|z|^{1/4}}\sin\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right)\]

Oscillatory behavior

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