Calculate Derivative Airy Function

Online calculator and formulas for computing the derivative Airy functions

Derivative Airy Function Calculator

Derivative Airy Functions

The derivative Airy functions Ai'(x) and Bi'(x) are special functions for solving linear differential equations y'' - xy = 0.

Real number as argument
Result
Ai'(x):
Bi'(x):

Derivative Airy Function Curves

Mouse pointer on the graph shows the values

Derivative Airy Function Formulas

Ai'(x) - Derivative of the First Kind

The derivative of the Airy function of the first kind is a solution to the Airy equation. It is frequently used in quantum mechanics, optics, and electromagnetism.

\[\displaystyle Ai'(x)=\frac{x}{\pi\sqrt{3}} K_{\frac{2}{3}}\left(\frac{2}{3}x^{\frac{3}{2}} \right)\]
Bi'(x) - Derivative of the Second Kind

The derivative of the Airy function of the second kind is another solution to the Airy equation. It is linearly independent of Ai'(x).

\[\displaystyle Bi'(x)= \frac{x}{\sqrt{3}} \left(I_{-\frac{2}{3}} \left(\frac{2}{3}x^{\frac{3}{2}}\right) + I_{\frac{2}{3}} \left(\frac{2}{3}x^{\frac{3}{2}} \right) \right)\]
Note

Here \(I\) is the modified Bessel function. These functions are of particular interest in mathematical physics and have diverse applications.

Detailed Description of the Derivative Airy Functions

Mathematical Foundations

The derivative Airy functions \(\displaystyle Ai'(x) \) and the related function \(\displaystyle Bi'(x)\) denote special functions in mathematics for solving the linear differential equation \(\displaystyle y'' -xy=0\).

Calculation Instructions

On this page, the derivative Airy function is calculated. To calculate, enter the argument, then click the 'Calculate' button.

Applications

Quantum Mechanics

The derivative Airy functions frequently appear in quantum mechanics, particularly in solving the Schrödinger equation for certain potentials.

Optics and Electromagnetism

In optics and electromagnetism, the derivative Airy functions are used to describe wave phenomena and radiation transfer.

Mathematical Physics

These functions are closely related to the Airy function and appear in various scientific contexts of mathematical physics.

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Airy • Derivative Airy • Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye • Spherical-Bessel-J • Spherical-Bessel-Y • Hankel • Beta • Incomplete Beta • Incomplete Inverse Beta • Binomial Coefficient • Binomial Coefficient Logarithm • Erf • Erfc • Erfi • Erfci • Fibonacci • Fibonacci Tabelle • Gamma • Inverse Gamma • Log Gamma • Digamma • Trigamma • Logit • Sigmoid • Derivative Sigmoid • Softsign • Derivative Softsign • Softmax • Struve • Struve table • Modified Struve • Modified Struve table • Riemann Zeta