airyBi¶
Evaluates the Airy function of the second kind.
Synopsis¶
airyBi (x)
Required Arguments¶
- float
x(Input) - Argument for which the function value is desired.
Return Value¶
The Airy function of the second kind evaluated at x, \(Bi(x)\).
Description¶
The airy function \(Bi(x)\) is defined to be
\[Bi(x) = \tfrac{1}{\pi} \int_0^{\infty} \exp \left(xt - \tfrac{1}{3} t^3\right)dt +
\tfrac{1}{\pi} \int_0^{\infty} \sin \left(xt + \tfrac{1}{3} t^3\right) dt\]
It can also be expressed in terms of modified Bessel functions of the first kind, \(I_v(x)\), and Bessel functions of the first kind \(J_v(x)\) (see besselIx and besselJx):
\[Bi(x) = \sqrt{\tfrac{x}{3}}
\left[I_{-1/3} \left(\tfrac{2}{3}x^{3/2}\right) +
I_{1/3}\left(\tfrac{2}{3}x^{3/2}\right)\right]
\text{ for } x > 0\]
and
\[Bi(x) = \sqrt{\tfrac{-x}{3}}
\left[J_{-1/3} \left(\tfrac{2}{3}|x|^{3/2}\right) -
J_{1/3}\left(\tfrac{2}{3}|x|^{3/2}\right)\right]
\text{ for } x < 0\]
Let ɛ = machine(4), the machine precision. If \(x<-1.31
\varepsilon^{-2/3}\), then the answer will have no precision.
If \(x<-1 31 \varepsilon^{-1/3}\), the answer will be less accurate than half precision. In addition, x should not be so large that \(\exp \left[(2/3) x^{3/2} \right]\) overflows. For more information, see the description for machine.