Airy functions#

This module implements Airy functions and their generalized derivatives. It supports symbolic functionality through Maxima and numeric evaluation through mpmath and scipy.

Airy functions are solutions to the differential equation \(f''(x) - x f(x) = 0\).

Four global function symbols are immediately available, please see

airy_ai(): for the Airy Ai function
airy_ai_prime(): for the first differential of the Airy Ai function
airy_bi(): for the Airy Bi function
airy_bi_prime(): for the first differential
of the Airy Bi function

AUTHORS:

Oscar Gerardo Lazo Arjona (2010): initial version
Douglas McNeil (2012): rewrite

EXAMPLES:

Verify that the Airy functions are solutions to the differential equation:

sage: diff(airy_ai(x), x, 2) - x * airy_ai(x)                                       # needs sage.symbolic
0
sage: diff(airy_bi(x), x, 2) - x * airy_bi(x)                                       # needs sage.symbolic
0

class sage.functions.airy.FunctionAiryAiGeneral#

Bases: BuiltinFunction

The generalized derivative of the Airy Ai function

INPUT:

alpha – Return the \(\alpha\)-th order fractional derivative with respect to \(z\). For \(\alpha = n = 1,2,3,\ldots\) this gives the derivative \(\operatorname{Ai}^{(n)}(z)\), and for \(\alpha = -n = -1,-2,-3,\ldots\) this gives the \(n\)-fold iterated integral.

\[ \begin{align}\begin{aligned}f_0(z) = \operatorname{Ai}(z)\\f_n(z) = \int_0^z f_{n-1}(t) dt\end{aligned}\end{align} \]

x – The argument of the function

EXAMPLES:

sage: # needs sage.symbolic
sage: from sage.functions.airy import airy_ai_general
sage: x, n = var('x n')
sage: airy_ai_general(-2, x)
airy_ai(-2, x)
sage: derivative(airy_ai_general(-2, x), x)
airy_ai(-1, x)
sage: airy_ai_general(n, x)
airy_ai(n, x)
sage: derivative(airy_ai_general(n, x), x)
airy_ai(n + 1, x)

class sage.functions.airy.FunctionAiryAiPrime#

Bases: BuiltinFunction

The derivative of the Airy Ai function; see airy_ai() for the full documentation.

EXAMPLES:

sage: # needs sage.symbolic
sage: x, n = var('x n')
sage: airy_ai_prime(x)
airy_ai_prime(x)
sage: airy_ai_prime(0)
-1/3*3^(2/3)/gamma(1/3)
sage: airy_ai_prime(x)._sympy_()                                            # needs sympy
airyaiprime(x)

class sage.functions.airy.FunctionAiryAiSimple#

Bases: BuiltinFunction

The class for the Airy Ai function.

EXAMPLES:

sage: from sage.functions.airy import airy_ai_simple
sage: f = airy_ai_simple(x); f                                              # needs sage.symbolic
airy_ai(x)
sage: airy_ai_simple(x)._sympy_()                                           # needs sage.symbolic
airyai(x)

class sage.functions.airy.FunctionAiryBiGeneral#

Bases: BuiltinFunction

The generalized derivative of the Airy Bi function.

INPUT:

alpha – Return the \(\alpha\)-th order fractional derivative with respect to \(z\). For \(\alpha = n = 1,2,3,\ldots\) this gives the derivative \(\operatorname{Bi}^{(n)}(z)\), and for \(\alpha = -n = -1,-2,-3,\ldots\) this gives the \(n\)-fold iterated integral.

\[ \begin{align}\begin{aligned}f_0(z) = \operatorname{Bi}(z)\\f_n(z) = \int_0^z f_{n-1}(t) dt\end{aligned}\end{align} \]

x – The argument of the function

EXAMPLES:

sage: # needs sage.symbolic
sage: from sage.functions.airy import airy_bi_general
sage: x, n = var('x n')
sage: airy_bi_general(-2, x)
airy_bi(-2, x)
sage: derivative(airy_bi_general(-2, x), x)
airy_bi(-1, x)
sage: airy_bi_general(n, x)
airy_bi(n, x)
sage: derivative(airy_bi_general(n, x), x)
airy_bi(n + 1, x)

class sage.functions.airy.FunctionAiryBiPrime#

Bases: BuiltinFunction

The derivative of the Airy Bi function; see airy_bi() for the full documentation.

EXAMPLES:

sage: # needs sage.symbolic
sage: x, n = var('x n')
sage: airy_bi_prime(x)
airy_bi_prime(x)
sage: airy_bi_prime(0)
3^(1/6)/gamma(1/3)
sage: airy_bi_prime(x)._sympy_()                                            # needs sympy
airybiprime(x)

class sage.functions.airy.FunctionAiryBiSimple#

Bases: BuiltinFunction

The class for the Airy Bi function.

EXAMPLES:

sage: from sage.functions.airy import airy_bi_simple
sage: f = airy_bi_simple(x); f                                              # needs sage.symbolic
airy_bi(x)
sage: f._sympy_()                                                           # needs sympy sage.symbolic
airybi(x)

sage.functions.airy.airy_ai(alpha, x=None, hold_derivative=True, **kwds)#

The Airy Ai function

The Airy Ai function \(\operatorname{Ai}(x)\) is (along with \(\operatorname{Bi}(x)\)) one of the two linearly independent standard solutions to the Airy differential equation \(f''(x) - x f(x) = 0\). It is defined by the initial conditions:

\[ \begin{align}\begin{aligned}\operatorname{Ai}(0)=\frac{1}{2^{2/3} \Gamma\left(\frac{2}{3}\right)},\\\operatorname{Ai}'(0)=-\frac{1}{2^{1/3}\Gamma\left(\frac{1}{3}\right)}.\end{aligned}\end{align} \]

Another way to define the Airy Ai function is:

\[\operatorname{Ai}(x)=\frac{1}{\pi}\int_0^\infty \cos\left(\frac{1}{3}t^3+xt\right) dt.\]

INPUT:

alpha – Return the \(\alpha\)-th order fractional derivative with respect to \(z\). For \(\alpha = n = 1,2,3,\ldots\) this gives the derivative \(\operatorname{Ai}^{(n)}(z)\), and for \(\alpha = -n = -1,-2,-3,\ldots\) this gives the \(n\)-fold iterated integral.

\[ \begin{align}\begin{aligned}f_0(z) = \operatorname{Ai}(z)\\f_n(z) = \int_0^z f_{n-1}(t) dt\end{aligned}\end{align} \]

x – The argument of the function
hold_derivative – Whether or not to stop from returning higher derivatives in terms of \(\operatorname{Ai}(x)\) and \(\operatorname{Ai}'(x)\)

See also

airy_bi()

EXAMPLES:

sage: n, x = var('n x')                                                         # needs sage.symbolic
sage: airy_ai(x)                                                                # needs sage.symbolic
airy_ai(x)

It can return derivatives or integrals:

sage: # needs sage.symbolic
sage: airy_ai(2, x)
airy_ai(2, x)
sage: airy_ai(1, x, hold_derivative=False)
airy_ai_prime(x)
sage: airy_ai(2, x, hold_derivative=False)
x*airy_ai(x)
sage: airy_ai(-2, x, hold_derivative=False)
airy_ai(-2, x)
sage: airy_ai(n, x)
airy_ai(n, x)

It can be evaluated symbolically or numerically for real or complex values:

sage: airy_ai(0)                                                                # needs sage.symbolic
1/3*3^(1/3)/gamma(2/3)
sage: airy_ai(0.0)                                                              # needs mpmath
0.355028053887817
sage: airy_ai(I)                                                                # needs sage.symbolic
airy_ai(I)
sage: airy_ai(1.0*I)                                                            # needs sage.symbolic
0.331493305432141 - 0.317449858968444*I

The functions can be evaluated numerically either using mpmath. which can compute the values to arbitrary precision, and scipy:

sage: airy_ai(2).n(prec=100)                                                    # needs sage.symbolic
0.034924130423274379135322080792
sage: airy_ai(2).n(algorithm='mpmath', prec=100)                                # needs sage.symbolic
0.034924130423274379135322080792
sage: airy_ai(2).n(algorithm='scipy')  # rel tol 1e-10                          # needs scipy sage.symbolic
0.03492413042327323

And the derivatives can be evaluated:

sage: airy_ai(1, 0)                                                             # needs sage.symbolic
-1/3*3^(2/3)/gamma(1/3)
sage: airy_ai(1, 0.0)                                                           # needs mpmath
-0.258819403792807

Plots:

sage: plot(airy_ai(x), (x, -10, 5)) + plot(airy_ai_prime(x),                    # needs sage.plot sage.symbolic
....:  (x, -10, 5), color='red')
Graphics object consisting of 2 graphics primitives

REFERENCES:

Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 10”
Wikipedia article Airy_function

sage.functions.airy.airy_bi(alpha, x=None, hold_derivative=True, **kwds)#

The Airy Bi function

The Airy Bi function \(\operatorname{Bi}(x)\) is (along with \(\operatorname{Ai}(x)\)) one of the two linearly independent standard solutions to the Airy differential equation \(f''(x) - x f(x) = 0\). It is defined by the initial conditions:

\[ \begin{align}\begin{aligned}\operatorname{Bi}(0)=\frac{1}{3^{1/6} \Gamma\left(\frac{2}{3}\right)},\\\operatorname{Bi}'(0)=\frac{3^{1/6}}{ \Gamma\left(\frac{1}{3}\right)}.\end{aligned}\end{align} \]

Another way to define the Airy Bi function is:

\[\operatorname{Bi}(x)=\frac{1}{\pi}\int_0^\infty \left[ \exp\left( xt -\frac{t^3}{3} \right) +\sin\left(xt + \frac{1}{3}t^3\right) \right ] dt.\]

INPUT:

alpha – Return the \(\alpha\)-th order fractional derivative with respect to \(z\). For \(\alpha = n = 1,2,3,\ldots\) this gives the derivative \(\operatorname{Bi}^{(n)}(z)\), and for \(\alpha = -n = -1,-2,-3,\ldots\) this gives the \(n\)-fold iterated integral.

\[ \begin{align}\begin{aligned}f_0(z) = \operatorname{Bi}(z)\\f_n(z) = \int_0^z f_{n-1}(t) dt\end{aligned}\end{align} \]

x – The argument of the function
hold_derivative – Whether or not to stop from returning higher derivatives in terms of \(\operatorname{Bi}(x)\) and \(\operatorname{Bi}'(x)\)