Derivative Airy Functions for Complex Numbers

Calculation of Ai'(z) and Bi'(z) - Derivatives of the Airy Functions

Derivative Airy Functions Calculator

Derivative Airy Functions

The derivative Airy functions Ai'(z) and Bi'(z) are the first derivatives of the Airy functions Ai(z) and Bi(z). They are also solutions of a related differential equation and play an important role in quantum mechanics and optics.

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

Derivative Airy - Properties

Relation to Airy Equation
\[\frac{d}{dz}Ai(z) = Ai'(z)\] \[\frac{d}{dz}Bi(z) = Bi'(z)\]

First derivatives of the Airy functions

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

Satisfied by Ai'(z) and Bi'(z)

Ai'(z) Derivative 1st Kind
Bi'(z) Derivative 2nd Kind
Important Properties
  • Ai'(0) ≈ -0.25881940...
  • Bi'(0) ≈ 0.44828835...
  • Ai'(z) → 0 for z → +∞
  • Bi'(z) → ∞ for z → +∞
Related Functions

Airy Functions:
Ai(z) and Bi(z) →

Real Numbers:
Ai'(x) and Bi'(x) for real numbers →

Applications
  • Quantum mechanics: WKB connection formulas
  • Optics: Intensity gradient
  • Electromagnetism: Field gradients
  • Mathematics: Asymptotic analysis

Formulas for the Derivative Airy Functions

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

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

With modified Bessel function \(K_{2/3}\)

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

With modified Bessel functions \(I_{\pm 2/3}\)

Derivative Airy Functions - Detailed Description

Definition

The derivative Airy functions are the first derivatives of the Airy functions Ai(z) and Bi(z).

Definitions:
\[Ai'(z) = \frac{d}{dz}Ai(z)\]
\[Bi'(z) = \frac{d}{dz}Bi(z)\]

They satisfy the differential equation:
\[y'' - zy = 0\]
where y = Ai'(z) or y = Bi'(z)

Ai'(z) - First Derivative

The derivative of the Airy function of the first kind:

Behavior:

  • Ai'(z) → 0 for z → +∞ (exponential decay)
  • Ai'(z) oscillates for z < 0
  • Ai'(0) ≈ -0.25881940...
  • Ai' has negative slope at z=0

Bi'(z) - Second Derivative

The derivative of the Airy function of the second kind:

Behavior:

  • Bi'(z) → ∞ for z → +∞ (exponential growth)
  • Bi'(z) oscillates for z < 0
  • Bi'(0) ≈ 0.44828835...
  • Bi' has positive slope at z=0

Wronskian Determinant

Wronskian for derivatives:
\[W = Ai'(z)Bi''(z) - Ai''(z)Bi'(z)\]
is also constant

Relations

Recurrence relations:
\(Ai''(z) = z \cdot Ai(z)\)
\(Bi''(z) = z \cdot Bi(z)\)

Linearly independent:
Ai'(z) and Bi'(z) are linearly independent

Physical Applications

Quantum Mechanics:
• WKB method: Connection formulas at turning points
• Tunneling effect: Wave function derivatives
• Potential problems: Boundary conditions
• Scattering theory: Phase shifts
Optics and Electromagnetism:
• Intensity gradient: Derivative of light intensity
• Field gradients: Electromagnetic fields
• Wave propagation: Phase velocity
• Caustics: Gradient at focal lines

Mathematical Properties

Zeros

• Ai'(z) has infinitely many zeros for z < 0
• Bi'(z) has infinitely many zeros for z < 0
• Important for eigenvalue problems

Asymptotics

• For z → +∞: exponential behavior
• For z → -∞: oscillatory behavior
• Similar to Ai(z) and Bi(z)

Integrals

• \(\int Ai'(z)dz = Ai(z) + C\)
• \(\int Bi'(z)dz = Bi(z) + C\)
• Antiderivatives are the Airy functions

Important Relations

To the Airy equation:

If y(z) satisfies the Airy equation \(y'' - zy = 0\),
then y'(z) satisfies the equation:
\((y')'' - zy' - y = 0\)

Integral representation:

\[Ai'(z) = -\frac{1}{\pi}\int_0^\infty t\sin\left(\frac{t^3}{3} + zt\right)dt\] Derivative of the integral representation of Ai(z)

Asymptotic Behavior

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

Exponential decay, negative sign

Ai'(z) for z → -∞
\[Ai'(z) \sim -\frac{|z|^{1/4}}{\sqrt{\pi}}\cos\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right)\]

Oscillatory behavior

Bi'(z) for z → +∞
\[Bi'(z) \sim \frac{z^{1/4}}{\sqrt{\pi}}e^{\frac{2}{3}z^{3/2}}\]

Exponential growth, positive sign

Bi'(z) for z → -∞
\[Bi'(z) \sim \frac{|z|^{1/4}}{\sqrt{\pi}}\sin\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right)\]

Oscillatory behavior with sine

Comparison with Airy Functions

The derivative functions have similar asymptotic behavior to the Airy functions themselves, but with an additional factor \(z^{1/4}\) or \(|z|^{1/4}\) and slightly shifted phases.


More complex functions

Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •
Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •