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Fungrim entry: c9bcf7

n=01(2nn)xn=2F1 ⁣(1,1,12,x4)=11u+uasin ⁣(u)(1u)3/2   where u=x4\sum_{n=0}^{\infty} \frac{1}{{2 n \choose n}} {x}^{n} = \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{x}{4}\right) = \frac{1}{1 - u} + \frac{\sqrt{u} \operatorname{asin}\!\left(\sqrt{u}\right)}{{\left(1 - u\right)}^{3 / 2}}\; \text{ where } u = \frac{x}{4}
Assumptions:xCandx<4x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| < 4
TeX:
\sum_{n=0}^{\infty} \frac{1}{{2 n \choose n}} {x}^{n} = \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{x}{4}\right) = \frac{1}{1 - u} + \frac{\sqrt{u} \operatorname{asin}\!\left(\sqrt{u}\right)}{{\left(1 - u\right)}^{3 / 2}}\; \text{ where } u = \frac{x}{4}

x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| < 4
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
Infinity\infty Positive infinity
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("c9bcf7"),
    Formula(Equal(Sum(Mul(Div(1, Binomial(Mul(2, n), n)), Pow(x, n)), Tuple(n, 0, Infinity)), Hypergeometric2F1(1, 1, Div(1, 2), Div(x, 4)), Where(Add(Div(1, Sub(1, u)), Div(Mul(Sqrt(u), Asin(Sqrt(u))), Pow(Sub(1, u), Div(3, 2)))), Equal(u, Div(x, 4))))),
    Variables(x),
    Assumptions(And(Element(x, CC), Less(Abs(x), 4))))

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

2019-08-21 11:44:15.926409 UTC