# ZernikeR

ZernikeR[n,m,r]

gives the radial Zernike polynomial .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• Explicit polynomials are given when possible.
• The Zernike polynomials are orthogonal with weight over the unit interval.
• ZernikeR can be evaluated to arbitrary numerical precision.
• ZernikeR automatically threads over lists.
• ZernikeR can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(4)

Evaluate numerically:

Evaluate symbolically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(26)

### Numerical Evaluation(6)

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:

Compute the elementwise values of an array:

Or compute the matrix ZernikeR function using MatrixFunction:

### Specific Values(3)

Values of ZernikeR at fixed points:

Value at zero:

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

### Visualization(3)

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:

### Function Properties(10)

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:

### Differentiation(2)

First derivative with respect to r:

Higher derivatives with respect to r:

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

### Function Identities and Simplifications(2)

ZernikeR is defined in terms of the Jacobi polynomial:

ZernikeR may reduce to a simpler form:

## Applications(1)

A function to convert a radial representation to a Cartesian one:

Visualize the combined effect of -astigmatism and -coma aberrations:

## Properties & Relations(6)

Obtain a sequence of Zernike polynomials from their generating function:

Compare with the directly computed sequence:

Verify the differential equation satisfied by the Zernike polynomial:

Verify recurrence relations satisfied by Zernike polynomials:

An integral representation of the radial Zernike polynomial:

Compare with the result of ZernikeR:

ZernikeR can be represented in terms of MeijerG:

Radial Zernike polynomials are orthogonal on the unit interval with weight function :

## Neat Examples(1)

A function for converting from OSA/ANSI standard indexing to Zernike polynomial indices:

Define the Zernike polynomial over the unit disk:

Visualize the first few Zernike polynomials:

Wolfram Research (2007), ZernikeR, Wolfram Language function, https://reference.wolfram.com/language/ref/ZernikeR.html.

#### Text

Wolfram Research (2007), ZernikeR, Wolfram Language function, https://reference.wolfram.com/language/ref/ZernikeR.html.

#### CMS

Wolfram Language. 2007. "ZernikeR." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ZernikeR.html.

#### APA

Wolfram Language. (2007). ZernikeR. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ZernikeR.html

#### BibTeX

@misc{reference.wolfram_2024_zerniker, author="Wolfram Research", title="{ZernikeR}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ZernikeR.html}", note=[Accessed: 13-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_zerniker, organization={Wolfram Research}, title={ZernikeR}, year={2007}, url={https://reference.wolfram.com/language/ref/ZernikeR.html}, note=[Accessed: 13-September-2024 ]}