${y}^{(n)}(z) = z {y}^{(n - 2)}(z) + \left(n - 2\right) {y}^{(n - 3)}(z)\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}$
TeX:
{y}^{(n)}(z) = z {y}^{(n - 2)}(z) + \left(n - 2\right) {y}^{(n - 3)}(z)\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
AiryAi$\operatorname{Ai}\!\left(z\right)$ Airy function of the first kind
AiryBi$\operatorname{Bi}\!\left(z\right)$ Airy function of the second kind
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("eadca2"),
Formula(Where(Equal(ComplexDerivative(y(z), For(z, z, n)), Add(Mul(z, ComplexDerivative(y(z), For(z, z, Sub(n, 2)))), Mul(Sub(n, 2), ComplexDerivative(y(z), For(z, z, Sub(n, 3)))))), Equal(y(z), Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z)))))),
Variables(n, z, C, D),
Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(3)), Element(C, CC), Element(D, CC))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC