Fungrim entry: bb5d67

z y''(z) + a y'(z) - y(z) = 0\; \text{ where } y(z) = C \,{}_0{\textbf F}_1\!\left(a, z\right) + D {z}^{1 - a} \,{}_0{\textbf F}_1\!\left(2 - a, z\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(z \ne 0 \;\mathbin{\operatorname{or}}\; 1 - a \in \mathbb{Z}_{\ge 0}\right)

TeX:

z y''(z) + a y'(z) - y(z) = 0\; \text{ where } y(z) = C \,{}_0{\textbf F}_1\!\left(a, z\right) + D {z}^{1 - a} \,{}_0{\textbf F}_1\!\left(2 - a, z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(z \ne 0 \;\mathbin{\operatorname{or}}\; 1 - a \in \mathbb{Z}_{\ge 0}\right)

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("bb5d67"),
    Formula(Where(Equal(Sub(Add(Mul(z, ComplexDerivative(y(z), For(z, z, 2))), Mul(a, ComplexDerivative(y(z), For(z, z, 1)))), y(z)), 0), Equal(y(z), Add(Mul(C, Hypergeometric0F1Regularized(a, z)), Mul(Mul(D, Pow(z, Sub(1, a))), Hypergeometric0F1Regularized(Sub(2, a), z)))))),
    Variables(z, a, C, D),
    Assumptions(And(Element(a, CC), Element(z, CC), Element(C, CC), Element(D, CC), Or(NotEqual(z, 0), Element(Sub(1, a), ZZGreaterEqual(0))))))

Topics using this entry

Confluent hypergeometric functions