The Special Function {{\rm Ai}\left(x\right)}


1. Differential Equation[-]

The function {{\rm Ai}\left(x\right)} satisfies the differential equation with initial values y \left( 0 \right) =\frac{\sqrt [3]{3}}{3\,\Gamma \left( 2/3 \right) } and y' \left( 0 \right) =-\frac{\sqrt [6]{3}\,\Gamma \left( 2/3 \right) }{2\pi }.

2. Plot[+]

3. Numerical Evaluation[+]

4. Derivative in Terms of Lower-Order Derivatives[+]

5. Expansion at 0[+]

6. Local Expansions at Singularities and at Infinity[-]

The differential equation above has 0 non-zero finite singular pointsA complex numberĀ z_0 is a singular point (or singularity of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation vanishes atĀ z_0..

6.1. Expansion at \infty [+]

7. Hypergeometric Representation[+]

8. Chebyshev Expansion over [-1,1][+]

9. Laplace Transform[+]