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:

    “There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
    —Adrienne Rich (b. 1929)

    “... a worker was seldom so much annoyed by what he got as by what he got in relation to his fellow workers.”
    —Mary Barnett Gilson (1877–?)

    “The rebellion is against time pollution, the feeling that the essence of what makes life worth living—the small moments, the special family getaways, the cookies in the oven, the weekend drives, the long dreamlike summers Mso much of this has been taken from us, or we have given it up. For what? Hitachi stereos? Club Med? Company cars? Racquetball? For fifteen-hour days and lousy day care?”
    —Richard Louv (20th century)

    “Mark the babe
    Not long accustomed to this breathing world;
    One that hath barely learned to shape a smile,
    Though yet irrational of soul, to grasp
    With tiny finger—to let fall a tear;
    And, as the heavy cloud of sleep dissolves,
    To stretch his limbs, bemocking, as might seem,
    The outward functions of intelligent man.”
    —William Wordsworth (1770–1850)