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Fungrim entry: b14da0

z+x=zk=0(1)k(12)kzkk!xk\sqrt{z + x} = \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}  and  xC  and  (x<z  and  (Re(z)>0  or  sgn ⁣(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}  and  x 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:
\sqrt{z + x} = \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
PowerSeriesK[[x]]K[[x]] Formal power series
Source code for this entry:
Entry(ID("b14da0"),
    Formula(Equal(Sqrt(Add(z, x)), Mul(Sqrt(z), Sum(Mul(Div(Mul(Pow(-1, k), RisingFactorial(Neg(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, PowerSeries(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.

2020-08-27 09:56:25.682319 UTC