Fungrim entry: 3c2557

\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:

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:

\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| \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("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)), 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

Square roots