# Fungrim entry: b14da0

$\sqrt{z + x} = \sqrt{z} \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k} \left(-\frac{1}{2}\right)_{k}}{{z}^{k} k !} {x}^{k}$
Assumptions:$z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left|x\right| \lt \left|z\right| \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}\!\left(z\right) \gt 0 \,\mathbin{\operatorname{or}}\, \operatorname{sgn}\!\left(\operatorname{Im}\!\left(x\right)\right) = \operatorname{sgn}\!\left(\operatorname{Im}\!\left(z\right)\right)\right)\right)$
Alternative assumptions:$z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \text{ is the generator of } \mathbb{C}[[x]]$
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| \lt \left|z\right| \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}\!\left(z\right) \gt 0 \,\mathbin{\operatorname{or}}\, \operatorname{sgn}\!\left(\operatorname{Im}\!\left(x\right)\right) = \operatorname{sgn}\!\left(\operatorname{Im}\!\left(z\right)\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
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Re$\operatorname{Re}\!\left(z\right)$ Real part
Sign$\operatorname{sgn}\!\left(z\right)$ Sign function
Im$\operatorname{Im}\!\left(z\right)$ Imaginary part
FormalPowerSeries$K[[x]]$ 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)), Tuple(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)))))

## Topics using this entry

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

2019-06-18 07:49:59.356594 UTC