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Fungrim entry: eadca2

y(n)(z)=zy(n2)(z)+(n2)y(n3)(z)   where y(z)=CAi ⁣(z)+DBi ⁣(z){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:zCandnZ3andCCandDCz \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
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
AiryAiAi ⁣(z)\operatorname{Ai}\!\left(z\right) Airy function of the first kind
AiryBiBi ⁣(z)\operatorname{Bi}\!\left(z\right) Airy function of the second kind
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\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.

2019-10-05 13:11:19.856591 UTC