# Fungrim entry: 06f229

$z y''(z) + \left(b - z\right) y'(z) - a y(z) = 0\; \text{ where } y(z) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(D = 0 \;\mathbin{\operatorname{or}}\; z \ne 0 \;\mathbin{\operatorname{or}}\; -a \in \mathbb{Z}_{\ge 0}\right)$
TeX:
z y''(z) + \left(b - z\right) y'(z) - a y(z) = 0\; \text{ where } y(z) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \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(D = 0 \;\mathbin{\operatorname{or}}\; z \ne 0 \;\mathbin{\operatorname{or}}\; -a \in \mathbb{Z}_{\ge 0}\right)
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Hypergeometric1F1Regularized$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$ Regularized Kummer confluent hypergeometric function
HypergeometricU$U\!\left(a, b, z\right)$ Tricomi confluent hypergeometric function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("06f229"),
Formula(Where(Equal(Sub(Add(Mul(z, ComplexDerivative(y(z), For(z, z, 2))), Mul(Sub(b, z), ComplexDerivative(y(z), For(z, z, 1)))), Mul(a, y(z))), 0), Equal(y(z), Add(Mul(C, Hypergeometric1F1Regularized(a, b, z)), Mul(D, HypergeometricU(a, b, z)))))),
Variables(z, a, b, C, D),
Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Element(C, CC), Element(D, CC), Or(Equal(D, 0), NotEqual(z, 0), Element(Neg(a), ZZGreaterEqual(0))))))

## Topics using this entry

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