Derivative Airy Function Calculator

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

To calculate, enter the argument, then click on the 'Calculate' button.

The Airy function here expects a real number as an argument. The Airy function for complex numbers can be found in the complex numbers section.

Formulas for the derived Airy functions

The derived Airy functions are special mathematical functions used in physics and mathematics. They are closely linked to the Airy function and appear in various scientific contexts. Here is some important information about the Airy derived functions:

\(Ai'(x)\): The derivative of the Airy function of the first kind is a solution to the Airy equation. It is often used in quantum mechanics, optics and electromagnetics. The formula for \(Ai'(x)\) is:

\(\displaystyle Ai'(x)=\frac{x}{π\sqrt{3}} K_{\frac{2}{3}}\left(\frac{2}{3}x^{\ frac{3}{2}} \right) \)

\(Bi'(x)\): The derivative of the Airy function of the second kind is another solution to the Airy equation. It is linearly independent of Ai'(x) and has the following formula:

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

Where \(I\) is the modified Bessel function.

These functions are of particular interest in mathematical physics and have diverse applications.