Airy function
Calculator and formulas for calculating the Airy functions for complex numbers
This function calculates the Airy functions Ai(x) and Bi(x) for complex numbers.
The Airy functionen \(\displaystyle Ai (x) \) and the related function \(\displaystyle Bi(x)\) denote a special function in mathematics for solving the linear differential equation\(\displaystyle y'' -xy=0\).
To perform the calculation, enter the complex number, then click the 'Calculate' button.
The Airy function for real numbers and function curves can be found here
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Formulas for Airy Functions
\(\displaystyle Ai(x)=\frac{1}{π}\sqrt{\frac{x}{3}}K_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}}\right)\)
\(\displaystyle Bi(x)=\sqrt{\frac{x}{3}} \left( I_{-\frac{1}{3}} \left( \frac{2}{3}x^{\frac{3}{2}} \right) + I_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}}\right)\right) \)
Description of the Airy function
The Airy function is a special mathematical function commonly found in physics and optics. It is named after the British astronomer George Biddell Airy, who used it in his work on optics. There are several variants of the Airy function, of which \(Ai(z)\) and \(Bi(z)\) are the most common.
\(Ai(z)\): The Airy function of the first kind is a solution of the Airy equation or also called Stokes equation. It occurs in optics, quantum mechanics, electromagnetics and radiation transfer.
\(Bi(z)\): The Airy function of the second kind is another solution to the Airy equation. It is linearly independent of \(Ai(z)\) and is also used in various physical contexts.
The Airy functions are closely related to the solution of the Schrödinger equation for a linear potential well. Their properties, zeros and asymptotic behavior are of particular interest.
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 •
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