Surd

Surd[x,n]

gives the real-valued n^(th) root of x.

Details

  • Surd[x,n] returns the real-valued n^(th) root of real-valued x for odd n.
  • Surd[x,n] returns the principal n^(th) 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 (RadicalBox[x, 2, MultilineFunction -> None, SurdForm -> True])=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 ^(th) 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 n^(th) 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:

Wolfram Research (2012), Surd, Wolfram Language function, https://reference.wolfram.com/language/ref/Surd.html.

Text

Wolfram Research (2012), Surd, Wolfram Language function, https://reference.wolfram.com/language/ref/Surd.html.

CMS

Wolfram Language. 2012. "Surd." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Surd.html.

APA

Wolfram Language. (2012). Surd. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Surd.html

BibTeX

@misc{reference.wolfram_2023_surd, author="Wolfram Research", title="{Surd}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/Surd.html}", note=[Accessed: 29-September-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_surd, organization={Wolfram Research}, title={Surd}, year={2012}, url={https://reference.wolfram.com/language/ref/Surd.html}, note=[Accessed: 29-September-2023 ]}