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:

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:

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 :

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 :

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:

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 :

General term in the series expansion using SeriesCoefficient:

The first-order Fourier series:

The Taylor expansion at a generic point: