Airy Function - Relation To Other Special Functions

Relation To Other Special Functions

For positive arguments, the Airy functions are related to the modified Bessel functions:

\begin{align} \mathrm{Ai}(x) &{}= \frac1\pi \sqrt{\frac13 x} \, K_{1/3}\left(\tfrac23 x^{3/2}\right), \\ \mathrm{Bi}(x) &{}= \sqrt{\frac13 x} \left(I_{1/3}\left(\tfrac23 x^{3/2}\right) + I_{-1/3}\left(\tfrac23 x^{3/2}\right)\right). \end{align}

Here, I±1/3 and K1/3 are solutions of

The first derivative of Airy function is

\mathrm{Ai'}(x) = - \frac{x} {\pi \sqrt{3}} \, K_{2/3}\left(\tfrac23 x^{3/2}\right) .

Functions and can be represented in terms of rapidly converged integrals (see also modified Bessel functions )

For negative arguments, the Airy function are related to the Bessel functions:

\begin{align} \mathrm{Ai}(-x) &{}= \frac13 \sqrt{x} \left(J_{1/3}\left(\tfrac23 x^{3/2}\right) + J_{-1/3}\left(\tfrac23 x^{3/2}\right)\right), \\ \mathrm{Bi}(-x) &{}= \sqrt{\frac13 x} \left(J_{-1/3}\left(\tfrac23 x^{3/2}\right) - J_{1/3}\left(\tfrac23 x^{3/2}\right)\right). \end{align}

Here, J±1/3 are solutions of .

The Scorer's functions solve the equation . They can also be expressed in terms of the Airy functions:

\begin{align} \mathrm{Gi}(x) &{}= \mathrm{Bi}(x) \int_x^\infty \mathrm{Ai}(t) \, dt + \mathrm{Ai}(x) \int_0^x \mathrm{Bi}(t) \, dt, \\ \mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt. \end{align}

Read more about this topic:  Airy Function

Famous quotes containing the words relation to, relation, special and/or functions:

    “Light is meaningful only in relation to darkness, and truth presupposes error. It is these mingled opposites which people our life, which make it pungent, intoxicating. We only exist in terms of this conflict, in the zone where black and white clash.”
    —Louis Aragon (1897–1982)

    “Art should exhilarate, and throw down the walls of circumstance on every side, awakening in the beholder the same sense of universal relation and power which the work evinced in the artist, and its highest effect is to make new artists.”
    —Ralph Waldo Emerson (1803–1882)

    “... there has been a very special man in my life for the past year. All I’ll say about him is that he’s kind, warm, mature, someone I can trust—and he’s not a politician.”
    —Donna Rice (b. c. 1962)

    “Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.”
    —Terri Apter (20th century)