Sqrt

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values of nested square roots

L. D. Servi: Nested Square Roots of 2 American Mathematical Monthly 110, 326-329 (2003)

L. D. Servi: Nested Square Roots of 2 American Mathematical Monthly 110, 326-329 (2003)

Values at infinities

General characteristics

Domain and analyticity

(Sqrt[z]) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function (Sqrt[z]) does not have poles and essential singularities.

Branch points

Branch cuts

Analytic continuations

(Sqrt[z]) is chosen to be the principal branch of the general square root function which has two sheets: ±(Sqrt[z]).

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions on branch cuts

Expansions at z==1

Expansions of (Sqrt[1+z]) at z==0

Asymptotic series expansions

Residue representations

Representations of (Sqrt[z])

Representations of (Sqrt[1+z])

Other series representations

Integral representations

Contour integral representations

Limit representations

P. W. Pedersen

Continued fraction representations

Andreas Lauschke (2006)

Andreas Lauschke (2006)

P. W. Pedersen

P. W. Pedersen

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Half-angle formulas

Multiple arguments

For products

For quotients

Power of arguments

Exponent of arguments

Some functions of arguments

Products, sums, and powers of the direct function

Products of the direct function

Powers of the direct function

Power of the product of the direct function

Nested direct and inverse functions

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

For the direct function itself

Definite integration

For the direct function itself

Involving the direct function

Involving related functions

Integral transforms

Laplace transforms

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

See power function.

Classical cases involving unit step UnitStep

See power function.

Through other functions

Representations through equivalent functions

With inverse function

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==(Sqrt[z]), which generically equal to z.

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(Sqrt[z]), which generically does not equal to z.

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(Sqrt[z]), which equal to z under restriction -(π/2)<Arg(z)≤(π/2).

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(Sqrt[z]), which equal to -z under restriction (π/2)<Arg(z)≤π∨-π<Arg(z)≤(π/2).

With related functions

Inequalities

Zeros

Theorems

Wigner's semi-circle law

The probability density p(x) for the eigenvalues of a unitary n×n Gaussian ensemble matrix is p(x)==θ(2n-(x^2))(1/π)(Sqrt[2n-(x^2)]).

The maximal power efficiency output of an endo-reversible Carnot heat engine operating in finite times with a linear Newtonian heat transfer

The maximal power efficiency η output of an endo-reversible Carnot heat engine operating in finite times with a linear Newtonian heat transfer is given by η=1-(Sqrt[(Subscript[T, c])/(Subscript[T, h])]) , where (Subscript[T, c]) and (Subscript[T, h]) are the cold and hot temperature of the heat reservoirs.

Differential-Algebraic Constants

To ensure the correctness of many formulas given in this collection over the whole complex plane, it is often necessary to work with expressions of the form (Sqrt[(z^2)])==(Sqrt[i z])(Sqrt[-i z]) , (Sqrt[(z^2)])/z, z(Sqrt[1/(z^2)]),(Sqrt[-(z^2)])==(Sqrt[z])(Sqrt[-z]), etc.. While in a textbook-mathematics setting these expressions are often simplified to z, ±1, 1, ±i z, etc, this cannot be done inside Mathematica. From a complex function point of view the Riemann surface of such functions are made from disconnected sheets. Inside Mathematica all branch cuts of all functions (that have branch cuts) follow uniquely from the branch cut of the power function (the logarithm function respectively). As a result the branch cuts related to functions such as (Sqrt[(z^2)]), (Sqrt[(z^2)])/z, z(Sqrt[1/(z^2)]), (Sqrt[-(z^2)])exist, although they do not start and end at branch points, but mostly extend from -i ∞ to i ∞ or from -∞ to ∞. 
In details we have:

(Subscript[ℬ𝒞, z])((Sqrt[(z^2)]))==(Subscript[ℬ𝒞, z])((Sqrt[i z])(Sqrt[-i z]))=={{(-i ∞,0),1},{(0,i ∞),-1}}

(lim_(ϵ->+0)) (Sqrt[(((x+ϵ))^2)])==- (Sqrt[(x^2)])/;i x>0

(lim_(ϵ->+0)) (Sqrt[(((x-ϵ))^2)])==- (Sqrt[(x^2)])/;i x<0

(Subscript[ℬ𝒞, z])((Sqrt[-(z^2)]))==(Subscript[ℬ𝒞, z])((Sqrt[z])(Sqrt[-z]))=={{(-∞,0),-i},{(0,∞),i}}

(lim_(ϵ->+0)) (Sqrt[-(((x-i ϵ))^2)])==- (Sqrt[-(x^2)])/;x<0

(lim_(ϵ->+0)) (Sqrt[-(((x+i ϵ))^2)])==- (Sqrt[-(x^2)])/;x>0

(Subscript[ℬ𝒞, z])(z(Sqrt[1/(z^2)]))=={{(-i ∞,0),-1},{(0,i ∞),1}}

(Subscript[ℬ𝒞, z])((Sqrt[z])(Sqrt[1/z]))=={{(-∞,0),{}}}

An expression of the form (Sqrt[(z^2)])/z , z(Sqrt[1/(z^2)])are called differential-algebraic constants because their derivative vanishes generically everywhere as a complex function (but not as a generalized function).

History

– Babylonians (1900 BC) gave the first approximate evaluations of (Sqrt[2]) and (Sqrt[ ])from other natural numbers
– H. Briggs (1617)
– I. Newton (1665) derived an infinite series expression for 1+z; the notation (Sqrt[z]) was introduced in 1675

The function sqrt is encountered often in mathematics and the natural sciences.

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