The function of Airy Have is one of the special Fonctions in mathematics, i.e. one of the remarkable functions frequently appearing in calculations. It bears the name of the British astronomer George Biddell Airy, which introduced it for its calculations of optics, in particular at the time of the study of the Arc-en-ciel. The function of Airy Have and the function Bi, which one calls function of Airy of second species, are solutions of the linear differential equation of a nature two

$y$ - xy = 0 \ qquad \,

known under the name of equation of Airy .

Definition

Function Have

The function of Airy is defined in all X real by the formula

$\backslash \; mathrm\; \{Have\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; pi\}\; \backslash \; int\_0^\; \backslash \; infty\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{t^3\}\; \{3\}\; +\; xt\; \backslash \; right)\; \backslash ,\; dt.$

who forms a semi-convergent integral (that can be proven by a Intégration by parts). A theorem of derivation of the integrals with parameters makes it possible to show that Have is solution of the equation of Airy

$y$ - xy = 0 \,

with for initial conditions

$$

\ mathrm {Have} (0) = \ frac {1} {3^ {2/3} \ Gamma (\ frac23)}, \ qquad \ mathrm {Have} '(0) = - \ frac {1} {3^ {1/3} \ Gamma (\ frac13)}.

The function has in particular a Point inflection in x=0 . In the field x>0 , Have ( X ) is positive, concave, and exponentially decrease towards 0. In the field x<0 , Have ( X ) oscillates around value 0 with an increasingly strong frequency and an increasingly low amplitude as - X grows. It is what the equivalents at the boundaries confirm (when X tends towards +∞)

$$

\ mathrm {Have} (X) \ sim \ frac {e^ {- \ frac23x^ {3/2}}} {2 \ sqrt \ pi \, x^ {1/4}} \ qquad \ mathrm {Have} (- X) \ sim \ frac {\ sin (\ frac23x^ {3/2} - \ frac14 \ pi)}{\ sqrt \ pi \, x^ {1/4}}. in which one sees appearing the function gamma.

Function of Airy of second species Bi

The solutions of the equation of Airy (others that the null solution) also have an oscillating behavior in the field x<0 . The function of Airy of second species, Bi, is the solution of the equation of Airy whose oscillations have even amplitude that those of Have in the vicinity of − ∞ and which presents a Déphasage π/2. She admits for equivalents at the boundaries

$$

\ mathrm {Bi} (X) \ sim \ frac {e^ {\ frac23x^ {3/2}}} {\ sqrt \ pi \, x^ {1/4}} \ qquad \ mathrm {Bi} (- X) \ sim \ frac {\ cos (\ frac23x^ {3/2} - \ frac14 \ pi)}{\ sqrt \ pi \, x^ {1/4}}.

The functions Have and Bi functions constitute a fundamental system of solutions of the equation of Airy, the second correspondent in the initial conditions

$$

\ mathrm {Bi} (0) = \ frac {1} {3^ {1/6} \ Gamma (\ frac23)}, \ qquad \ mathrm {Bi} '(0) = \ frac {3^ {1/6}} {\ Gamma (\ frac13)}.

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