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Fungrim entry: 06f229

zy(z)+(bz)y(z)ay ⁣(z)=0   where y ⁣(z)=C1F1 ⁣(a,b,z)+DU ⁣(a,b,z)z y''(z) + \left(b - z\right) y'(z) - a y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)
Assumptions:aCandbCandzCandCCandDCand(D=0orz0oraZ0)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\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = 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
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
Hypergeometric1F1Regularized1F1 ⁣(a,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) Regularized Kummer confluent hypergeometric function
HypergeometricUU ⁣(a,b,z)U\!\left(a, b, z\right) Tricomi confluent hypergeometric function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\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, Derivative(y(z), Tuple(z, z, 2))), Mul(Sub(b, z), Derivative(y(z), Tuple(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), Unequal(z, 0), Element(Neg(a), ZZGreaterEqual(0))))))

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

2019-08-25 15:30:03.056001 UTC