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Fungrim entry: 3c2557

1z+x=1zk=0(1)k(12)kzkk!xk\frac{1}{\sqrt{z + x}} = \frac{1}{\sqrt{z}} \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k} \left(\frac{1}{2}\right)_{k}}{{z}^{k} k !} {x}^{k}
Assumptions:zC{0}andxCand(x<zand(Re(z)>0orsgn ⁣(Im(x))=sgn ⁣(Im(z))))z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left|x\right| < \left|z\right| \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}(z) > 0 \,\mathbin{\operatorname{or}}\, \operatorname{sgn}\!\left(\operatorname{Im}(x)\right) = \operatorname{sgn}\!\left(\operatorname{Im}(z)\right)\right)\right)
Alternative assumptions:zC{0}andx is the generator of C[[x]]z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \text{ is the generator of } \mathbb{C}[[x]]
TeX:
\frac{1}{\sqrt{z + x}} = \frac{1}{\sqrt{z}} \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k} \left(\frac{1}{2}\right)_{k}}{{z}^{k} k !} {x}^{k}

z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left|x\right| < \left|z\right| \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}(z) > 0 \,\mathbin{\operatorname{or}}\, \operatorname{sgn}\!\left(\operatorname{Im}(x)\right) = \operatorname{sgn}\!\left(\operatorname{Im}(z)\right)\right)\right)

z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \text{ is the generator of } \mathbb{C}[[x]]
Definitions:
Fungrim symbol Notation Short description
Sqrtz\sqrt{z} Principal square root
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Factorialn!n ! Factorial
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
ReRe(z)\operatorname{Re}(z) Real part
Signsgn(z)\operatorname{sgn}(z) Sign function
ImIm(z)\operatorname{Im}(z) Imaginary part
FormalPowerSeriesK[[x]]K[[x]] Formal power series
Source code for this entry:
Entry(ID("3c2557"),
    Formula(Equal(Div(1, Sqrt(Add(z, x))), Mul(Div(1, Sqrt(z)), Sum(Mul(Div(Mul(Pow(-1, k), RisingFactorial(Div(1, 2), k)), Mul(Pow(z, k), Factorial(k))), Pow(x, k)), For(k, 0, Infinity))))),
    Variables(z, x),
    Assumptions(And(Element(z, SetMinus(CC, Set(0))), Element(x, CC), And(Less(Abs(x), Abs(z)), Or(Greater(Re(z), 0), Equal(Sign(Im(x)), Sign(Im(z)))))), And(Element(z, SetMinus(CC, Set(0))), FormalGenerator(x, FormalPowerSeries(CC, x)))))

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

2019-10-05 13:11:19.856591 UTC