Sqrt

Sqrt[z]

or gives the square root of z.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • can be entered using or (@z).
  • Sqrt[z] is converted to .
  • 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 .
  • z can also be used for input. The character is entered as sqrt or \[Sqrt].

Examples

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Basic Examples  (6)

Evaluate numerically:

Enter using :

Negative numbers have imaginary square roots:

Plot over a subset of the reals:

Plot over a subset of the complexes:

is not necessarily equal to :

It can be simplified to if one assumes :

Scope  (32)

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 realvalued 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 x for which Sqrt[x ]=2.1 using Solve:

Substitute in the result:

Visualize the result:

Visualization  (4)

Plot the real and imaginary parts of the Sqrt[x] function:

Compare the real and imaginary parts of and (Surd[x,2]):

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (4)

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 sqrt or \[Sqrt], followed by a number:

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 ^(th) 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 :

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:

is not automatically replaced by :

It can be simplified to if one assumes :

It can be simplified to TemplateBox[{x}, Abs] if one assumes x in TemplateBox[{}, Reals]:

PowerExpand can be used to force cancellation without assumptions:

Expand assuming real variables x and y:

Applications  (2)

Roots of a quadratic polynomial:

Generate periodic continued fractions:

Properties & Relations  (11)

Reduce combinations of square roots:

Evaluate power series involving square roots:

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:

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:

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