# 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

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## 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:

Surd threads elementwise over lists and matrices:

Surd can be used with Interval and CenteredInterval objects:

### 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:

## 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: