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Fungrim entry: 2df3e3

0F1 ⁣(a,z)=e2z1F1 ⁣(a12,2a1,4z)\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)
Assumptions:aC  and  zC  and  2a{1,0,}a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 a \notin \{1, 0, \ldots\}
\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; 2 a \notin \{1, 0, \ldots\}
Fungrim symbol Notation Short description
Hypergeometric0F10F1 ⁣(a,z)\,{}_0F_1\!\left(a, z\right) Confluent hypergeometric limit function
Expez{e}^{z} Exponential function
Sqrtz\sqrt{z} Principal square root
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(Hypergeometric0F1(a, z), Mul(Exp(Neg(Mul(2, Sqrt(z)))), Hypergeometric1F1(Sub(a, Div(1, 2)), Sub(Mul(2, a), 1), Mul(4, Sqrt(z)))))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC), NotElement(Mul(2, a), ZZLessEqual(1)))))

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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