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:

    “You see, I am alive, I am alive
    I stand in good relation to the earth
    I stand in good relation to the gods
    I stand in good relation to all that is beautiful
    I stand in good relation to the daughter of Tsen-tainte
    You see, I am alive, I am alive”
    —N. Scott Momaday (b. 1934)

    “Whoever has a keen eye for profits, is blind in relation to his craft.”
    —Sophocles (497–406/5 B.C.)

    “... 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)

    “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)