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:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Sqrt function using MatrixFunction:

Compute average-case statistical intervals using Around:

Values of Sqrt at fixed points:

Values at zero:

Values at infinity:

Find a value of for which using Solve:

Substitute in the result:

Visualize the result:

Plot the real and imaginary parts of the Sqrt 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 :

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:

is not an analytic function:

Nor is it meromorphic:

is neither non-decreasing nor non-increasing:

However, it is increasing where it is real valued:

is injective:

Not surjective:

is non-negative on its domain of definition:

has a branch cut singularity for :

However, it is continuous at the origin:

is neither convex nor concave:

However, it is concave where it is real valued:

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the derivative with respect to z:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

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:

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 if one assumes :

PowerExpand can be used to force cancellation without assumptions:

Expand assuming real variables x and y: