15.3 Airy Functions

The Airy functions $\mathrm{A}\mathrm{i}(x)$ and $\mathrm{B}\mathrm{i}(x)$ are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Section 10.4 and DLMF 9.

The Airy differential equation is:

The numerically satisfactory pair of solutions (DLMF 9.2#T1) on the real line are $y=\mathrm{A}\mathrm{i}(x)$ and $y=\mathrm{B}\mathrm{i}(x).$

These two solutions are oscillatory for $x<0$. $\mathrm{A}\mathrm{i}(x)$ is the solution subject to the condition that $y\to 0$ as $x\to +\mathrm{\infty },$ and $\mathrm{B}\mathrm{i}(x)$ is the second solution with the same amplitude as $\mathrm{A}\mathrm{i}(x)$ as $x\to -\mathrm{\infty }$ which differs in phase by $\pi /2.$ Also, $\mathrm{B}\mathrm{i}(x)$ is unbounded as $x\to +\mathrm{\infty }.$

If the argument $x$ is a real or complex floating point number, the numerical value of the function is returned.

Function: airy_ai (x) ¶

The Airy function $\mathrm{A}\mathrm{i}(x).$ See A&S eqn 10.4.2 and DLMF 9.

See also airy_bi, airy_dai, and airy_dbi.

Categories: Airy functions · Special functions ·

Function: airy_dai (x) ¶

The derivative of the Airy function $\mathrm{A}\mathrm{i}(x)$ :

$$\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{y}\mathrm{\_}\mathrm{d}\mathrm{a}\mathrm{i}(x)=\frac{d}{dx}\mathrm{A}\mathrm{i}(x)$$

See airy_ai.

Categories: Airy functions · Special functions ·

Function: airy_bi (x) ¶

The Airy function $\mathrm{B}\mathrm{i}(x)$ . See A&S eqn 10.4.3 and DLMF 9.

See airy_ai, and airy_dbi.

Categories: Airy functions · Special functions ·

Function: airy_dbi (x) ¶

The derivative of the Airy function $\mathrm{B}\mathrm{i}(x)$ :

$$\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{y}\mathrm{\_}\mathrm{d}\mathrm{b}\mathrm{i}(x)=\frac{d}{dx}\mathrm{B}\mathrm{i}(x)$$

See airy_ai, and airy_bi.

Categories: Airy functions · Special functions ·