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

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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[+]