Get the square root of an expression—Wolfram Documentation (2024)

Table of Contents
Basic Examples(6) Scope(38) Numerical Evaluation(6) Specific Values(4) Visualization(4) Function Properties(10) Differentiation(3) Integration(3) Series Expansions(4) Function Identities and Simplifications(4) Applications(4) Properties & Relations(12) Possible Issues(3) Neat Examples(2) References

Sqrt[z]

or Get the square root of an expression—Wolfram Documentation (1) gives the square root of z.

Get the square root of an expression—Wolfram Documentation (2)

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Get the square root of an expression—Wolfram Documentation (3) can be entered using Get the square root of an expression—Wolfram Documentation (4) or ∖(∖@z∖).
  • Sqrt[z] is converted to Get the square root of an expression—Wolfram Documentation (5).
  • Sqrt[z^2] is not automatically converted to z.
  • Sqrt[a b] is not automatically converted to Sqrt[a]Sqrt[b].
  • These conversions can be done using PowerExpand, but will typically be correct only for positive real arguments.
  • For certain special arguments, Sqrt automatically evaluates to exact values.
  • Sqrt can be evaluated to arbitrary numerical precision.
  • Sqrt automatically threads over lists.
  • In StandardForm, Sqrt[z] is printed as Get the square root of an expression—Wolfram Documentation (6).
  • √z can also be used for input. The √ character is entered as Get the square root of an expression—Wolfram Documentation (7)sqrtGet the square root of an expression—Wolfram Documentation (8) or \[Sqrt].
open allclose all

Basic Examples(6)

Evaluate numerically:

Enter Get the square root of an expression—Wolfram Documentation (9) using Get the square root of an expression—Wolfram Documentation (10):

Negative numbers have imaginary square roots:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Get the square root of an expression—Wolfram Documentation (11) is not necessarily equal to Get the square root of an expression—Wolfram Documentation (12):

It can be simplified to Get the square root of an expression—Wolfram Documentation (13) if one assumes Get the square root of an expression—Wolfram Documentation (14):

Scope(38)

Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Sqrt can deal with real‐valued intervals:

Sqrt threads elementwise over lists and matrices:

Specific Values(4)

Values of Sqrt at fixed points:

Values at zero:

Values at infinity:

Find a value of Get the square root of an expression—Wolfram Documentation (15) for which Get the square root of an expression—Wolfram Documentation (16) using Solve:

Substitute in the result:

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Visualize the result:

Visualization(4)

Plot the real and imaginary parts of the Sqrt function:

Compare the real and imaginary parts of Get the square root of an expression—Wolfram Documentation (17) and Get the square root of an expression—Wolfram Documentation (18) (Surd[x,2]):

Plot the real part of Get the square root of an expression—Wolfram Documentation (19):

Plot the imaginary part of Get the square root of an expression—Wolfram Documentation (20):

Polar plot with Get the square root of an expression—Wolfram Documentation (21):

Function Properties(10)

The real domain of Sqrt:

It is defined for all complex values:

Sqrt achieves all non-negative values on the reals:

The range for complex values is the right half-plane, excluding the negative imaginary axis:

Find limits at branch cuts:

Enter a √ character as Get the square root of an expression—Wolfram Documentation (22)sqrtGet the square root of an expression—Wolfram Documentation (23) or \[Sqrt], followed by a number:

Get the square root of an expression—Wolfram Documentation (24) is not an analytic function:

Nor is it meromorphic:

Get the square root of an expression—Wolfram Documentation (25) is neither non-decreasing nor non-increasing:

However, it is increasing where it is real valued:

Get the square root of an expression—Wolfram Documentation (26) is injective:

Not surjective:

Get the square root of an expression—Wolfram Documentation (27) is non-negative on its domain of definition:

Get the square root of an expression—Wolfram Documentation (28) has a branch cut singularity for Get the square root of an expression—Wolfram Documentation (29):

However, it is continuous at the origin:

Get the square root of an expression—Wolfram Documentation (30) is neither convex nor concave:

However, it is concave where it is real valued:

Differentiation(3)

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the Get the square root of an expression—Wolfram Documentation (31)Get the square root of an expression—Wolfram Documentation (32) derivative with respect to z:

Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around Get the square root of an expression—Wolfram Documentation (33):

The general term in the series expansion using SeriesCoefficient:

The first-order Fourier series:

The Taylor expansion at a generic point:

Function Identities and Simplifications(4)

Primary definition:

Connection with Exp and Log:

Get the square root of an expression—Wolfram Documentation (34) is not automatically replaced by Get the square root of an expression—Wolfram Documentation (35):

It can be simplified to Get the square root of an expression—Wolfram Documentation (36) if one assumes Get the square root of an expression—Wolfram Documentation (37):

It can be simplified to Get the square root of an expression—Wolfram Documentation (38) if one assumes Get the square root of an expression—Wolfram Documentation (39):

PowerExpand can be used to force cancellation without assumptions:

Expand assuming real variables x and y:

Applications(4)

Roots of a quadratic polynomial:

Generate periodic continued fractions:

Solve a differential equation with Sqrt:

Compute an elliptic integral from the Sqrt function:

Properties & Relations(12)

Sqrt[x] and Surd[x,2] are the same for non-negative real values:

For negative reals, Sqrt gives an imaginary result, whereas the real-valued Surd reports an error:

Get the square root of an expression—Wolfram Documentation (40)

Reduce combinations of square roots:

Evaluate power series involving square roots:

Expand a complex square root assuming variables are real valued:

Factor polynomials with square roots in coefficients:

Simplify handles expressions involving square roots:

There are many subtle issues in handling square roots for arbitrary complex arguments:

PowerExpand expands forms involving square roots:

It generically assumes that all variables are positive:

Finite sums of integers and square roots of integers are algebraic numbers:

Take limits accounting for branch cuts:

Sqrt can be represented as a DifferentialRoot:

The generating function for Sqrt:

Possible Issues(3)

Square root is discontinuous across its branch cut along the negative real axis:

Sqrt[x^2] cannot automatically be reduced to x:

With x assumed positive, the simplification can be done:

Use PowerExpand to do the formal reduction:

Along the branch cut, these are not the same:

Neat Examples(2)

Approximation to GoldenRatio:

Riemann surface for square root:

Power CubeRoot Surd PowerExpand SqrtBox

Characters: \[Sqrt]

    ▪
  • Some Mathematical Functions
    • ▪
  • Operators
    • ▪
  • Typing Square Roots
    ▪
  • Arithmetic Functions
    • ▪
  • Elementary Functions
    • ▪
    • ▪
  • Mathematical Functions

Introduced in 1988 (1.0) | Updated in 1996 (3.0)

Wolfram Research (1988), Sqrt, Wolfram Language function, https://reference.wolfram.com/language/ref/Sqrt.html (updated 1996).

Text

Wolfram Research (1988), Sqrt, Wolfram Language function, https://reference.wolfram.com/language/ref/Sqrt.html (updated 1996).

CMS

Wolfram Language. 1988. "Sqrt." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Sqrt.html.

APA

Wolfram Language. (1988). Sqrt. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sqrt.html

BibTeX

@misc{reference.wolfram_2024_sqrt, author="Wolfram Research", title="{Sqrt}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Sqrt.html}", note=[Accessed: 21-June-2024]}

BibLaTeX

@online{reference.wolfram_2024_sqrt, organization={Wolfram Research}, title={Sqrt}, year={1996}, url={https://reference.wolfram.com/language/ref/Sqrt.html}, note=[Accessed: 21-June-2024]}

Get the square root of an expression—Wolfram Documentation (2024)

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