Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

Surd

Surd[x,n]

gives the real-valued root of x.

Details

Surd[x,n] returns the real-valued root of real-valued x for odd n.
Surd[x,n] returns the principal root for non-negative real-valued x and even n.
For symbolic x in Surd[x,n], x is assumed to be real valued.
Surd can be evaluated to arbitrary numerical precision.
Surd automatically threads over lists. »
In StandardForm, Surd[x,n] formats as .
can be entered as surd, and moves between the fields.
Surd can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples (5)

Surd gives a real-valued root:

Plot over a subset of the reals:

Enter using surd, then use :

Note that this is not the same as , which is Power[x,1/3]:

Compare the real and imaginary parts of and over the reals:

Series expansion:

Scope (31)

Numerical Evaluation (5)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Surd function using MatrixFunction:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Specific Values (4)

Values at fixed points:

Evaluate symbolically:

Values at infinity:

Find a value of x for which ()=1.5:

Substitute in the result:

Visualize the result:

Visualization (4)

Plot the Surd function for various orders:

Visualize the absolute value and argument (sign) of for odd n:

The function has the same absolute value but a different argument for :

Compare the real and imaginary parts of and for even n:

Polar plot with :

Function Properties (8)

Surd[x,n] is defined for all real x when n is a positive, odd integer:

For positive, even n, it is defined for non-negative x:

For negative n, 0 is removed from the domain:

Surd is not defined for nonreal complex values:

Surd[x,n] achieves all non-negative real values when n is a positive even integer:

For positive odd n, its range is the whole real line:

For negative n, 0 is removed from the range:

Surd[x,n] is not an analytic function of x for any integer n:

is increasing for positive :

Decreasing for negative even :

Indefinite for negative odd :

is injective for :

Visualize for :

And it is surjective onto for odd, positive , but not other values of :

Visualize for :

has indefinite sign for odd :

It is non-negative on its real domain for even :

in general has both singularities and discontinuities at zero:

However, for positive odd it is continuous at the origin:

is neither convex nor concave for odd :

On its domain of definition, it is concave for positive even and convex of negative even :

Differentiation (3)

The first derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x:

Formula for the derivative with respect to x:

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 :

General term in the series expansion using SeriesCoefficient:

The first-order Fourier series:

The Taylor expansion at a generic point:

Applications (1)

With , the real vector field corresponding to the complex function is , and the trajectories that follow the field satisfy the differential equation . The implicit solution is for real , which corresponds to a family of circles that are tangent to the real axis at the origin:

In polar coordinates, the trajectories are for any real :

More generally, for where is an integer, the streamlines follow for constant :

This also works for negative powers:

For odd powers, care must be taken to ensure the first argument to Surd is non-negative:

Properties & Relations (3)

Surd[x,n] is only defined for real x and integer n:

Surd[x,n] is a bijection onto its domain of definition for every nonzero integer n:

Use Surd[x,n] to find the real root:

Use Power[x,1/n] or to find the principle complex root:

Possible Issues (1)

On the negative real axis, Surd[x,n] is undefined for even n:

On the negative real axis, Surd[x,n] is different from the principal root returned by Power[x,1/n]:

Neat Examples (1)

Plot a composition of Surd:

Top