Fungrim symbol: Sqrt

Symbol: Sqrt —

\sqrt{z}

— Principal square root

The principal square root

\sqrt{z}

is a function of one complex variable

z

It has a branch point singularity at

z = 0

and a branch cut on

\left(-\infty, 0\right]

where the value on

\left(-\infty, 0\right)

is taken to be continuous with the upper half plane.

The following table lists all conditions such that Sqrt(z) is defined in Fungrim.

Domain	Codomain
Numbers
$z \in \left[0, \infty\right)$	$\sqrt{z} \in \left[0, \infty\right)$
$z \in \mathbb{C}$	$\sqrt{z} \in \mathbb{C}$
Infinities
$z \in \left\{{\tilde \infty}\right\}$	$\sqrt{z} \in \left\{{\tilde \infty}\right\}$
$z \in \left\{ {e}^{i \theta} \infty : \theta \in \left(-\pi, \pi\right] \right\}$	$\sqrt{z} \in \left\{ {e}^{i \theta} \infty : \theta \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right] \right\}$
Formal power series
$z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \in \left(0, \infty\right)$	$\sqrt{z} \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \ne 0$	$\sqrt{z} \in \mathbb{C}[[x]]$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`ConstPi`	$\pi$	The constant pi (3.14...)
`FormalPowerSeries`	$K[[x]]$	Formal power series
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("21d9b8"),
    SymbolDefinition(Sqrt, Sqrt(z), "Principal square root"),
    Description("The principal square root", Sqrt(z), "is a function of one complex variable", z, "."),
    Description("It has a branch point singularity at", Equal(z, 0), "and a branch cut on", OpenClosedInterval(Neg(Infinity), 0), "where the value on", OpenInterval(Neg(Infinity), 0), "is taken to be continuous with the upper half plane."),
    Description("The following table lists all conditions such that", SourceForm(Sqrt(z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, ClosedOpenInterval(0, Infinity)), Element(Sqrt(z), ClosedOpenInterval(0, Infinity))), Tuple(Element(z, CC), Element(Sqrt(z), CC)), TableSection("Infinities"), Tuple(Element(z, Set(UnsignedInfinity)), Element(Sqrt(z), Set(UnsignedInfinity))), Tuple(Element(z, SetBuilder(Mul(Exp(Mul(ConstI, theta)), Infinity), theta, Element(theta, OpenClosedInterval(Neg(ConstPi), ConstPi)))), Element(Sqrt(z), SetBuilder(Mul(Exp(Mul(ConstI, theta)), Infinity), theta, Element(theta, OpenClosedInterval(Div(Neg(ConstPi), 2), Div(ConstPi, 2)))))), TableSection("Formal power series"), Tuple(And(Element(z, FormalPowerSeries(RR, x)), Element(SeriesCoefficient(z, x, 0), OpenInterval(0, Infinity))), And(Element(Sqrt(z), FormalPowerSeries(RR, x)))), Tuple(And(Element(z, FormalPowerSeries(CC, x)), Unequal(SeriesCoefficient(z, x, 0), 0)), And(Element(Sqrt(z), FormalPowerSeries(CC, x)))))))

Topics using this symbol

The symbol Sqrt appears in 26 topics:

Square roots (in 46 entries)
Inverse tangent (in 16 entries)
Chebyshev polynomials (in 12 entries)
Modular j-invariant (in 10 entries)
Error functions (in 9 entries)
Lambert W-function (in 7 entries)
Factorials and binomial coefficients (in 7 entries)
Specific values of Bessel functions (in 6 entries)
Sine (in 6 entries)
Legendre polynomials (in 6 entries)
Confluent hypergeometric functions (in 5 entries)
Bounds and inequalities for the gamma function (in 5 entries)
Partition function (in 3 entries)
Gamma function (in 3 entries)
Powers (in 2 entries)
Pi (in 2 entries)
Coulomb wave functions (in 2 entries)
Bernoulli numbers and polynomials (in 2 entries)
Riemann zeta function (in 1 entries)
Prime numbers (in 1 entries)
Natural logarithm (in 1 entries)
Modular transformations (in 1 entries)
Golden ratio (in 1 entries)
Gauss hypergeometric function (in 1 entries)
Definite integrals (in 1 entries)
Complex parts (in 1 entries)

Entries using this symbol

The symbol Sqrt appears in 156 entries: