Fungrim entry: eadca2

{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))))

Topics using this entry

Airy functions