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Airy Functions

Source: R/airy_functions.R
airy_functions.Rd

Functions to compute the Airy functions \(Ai\) and \(Bi\), their derivatives, and their zeros.

The Airy functions are the two linearly independent solutions to the differential equation:

$$y'' - xy = 0$$

Airy \(Ai\) Function: The first solution to the Airy differential equation. For negative \(x\) values, \(Ai(x)\) exhibits oscillatory behavior. For positive \(x\) values, \(Ai(x)\) monotonically decreases toward zero.

Airy \(Bi\) Function: The second solution to the Airy differential equation. For negative \(x\) values, \(Bi(x)\) exhibits cyclic oscillation. For positive \(x\) values, \(Bi(x)\) tends toward infinity.

Airy \(Ai'\) Function: The derivative of the first solution to the Airy differential equation. For negative \(x\) values, \(Ai'(x)\) displays cyclic oscillation. For positive \(x\) values, \(Ai'(x)\) approaches zero asymptotically.

Airy \(Bi'\) Function: The derivative of the second solution to the Airy differential equation. For negative \(x\) values, \(Bi'(x)\) oscillates cyclically. For positive \(x\) values, \(Bi'(x)\) increases toward infinity.

Zeros of Airy Functions: The zeros are the values where \(Ai(x) = 0\) or \(Bi(x) = 0\). The zeros are indexed starting from 1. The first few zeros are approximately:

  • \(Ai\): -2.33811, -4.08795, -5.52056, ...

  • \(Bi\): -1.17371, -3.27109, -4.83074, ...

All functions are implemented using relationships to Bessel functions for numerical accuracy.

Usage

airy_ai(x)

airy_bi(x)

airy_ai_prime(x)

airy_bi_prime(x)

airy_ai_zero(m = NULL, start_index = NULL, number_of_zeros = NULL)

airy_bi_zero(m = NULL, start_index = NULL, number_of_zeros = NULL)

Arguments

x

Input numeric value

m

The index of the zero to find (1-based indexing, so m=1 returns the first zero).

start_index

The starting index for the zeros (1-based).

number_of_zeros

The number of zeros to find.

Value

Single numeric value for the Airy functions and their derivatives, or a vector of length number_of_zeros for the multiple zero functions.

See also

Boost Documentation for more details on the mathematical background.

Examples

airy_ai(2)
#> [1] 0.03492413
airy_bi(2)
#> [1] 3.298095
airy_ai_prime(2)
#> [1] -0.05309038
airy_bi_prime(2)
#> [1] 4.100682
airy_ai_zero(1)
#> [1] -2.338107
airy_bi_zero(1)
#> [1] -1.173713
airy_ai_zero(start_index = 1, number_of_zeros = 5)
#> [1] -2.338107 -4.087949 -5.520560 -6.786708 -7.944134
airy_bi_zero(start_index = 1, number_of_zeros = 5)
#> [1] -1.173713 -3.271093 -4.830738 -6.169852 -7.376762