Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

ZernikeR can be used with Interval and CenteredInterval objects:

Values of ZernikeR at fixed points:

Value at zero:

Find the first positive minimum of ZernikeR[7,5,x ]:

Compute the associated ZernikeR[7,3,x] polynomial:

Different ZernikeR types give different symbolic forms:

Plot the ZernikeR function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Domain of ZernikeR of integer orders:

The range for ZernikeR of integer orders:

The range for complex values:

ZernikeR has the mirror property :

ZernikeR is an analytic function of x:

ZernikeR is neither non-decreasing nor non-increasing:

ZernikeR is not injective:

But surjective:

ZernikeR is neither non-negative nor non-positive:

ZernikeR has no singularities or discontinuities:

ZernikeR is neither convex nor concave:

TraditionalForm formatting:

First derivative with respect to r:

Higher derivatives with respect to r:

Plot the absolute values of the higher derivatives with respect to r:

ZernikeR is defined through the identity:

ZernikeR may reduce to simpler form: