ZernikeR

ZernikeR[n,m,r]

gives the radial Zernike polynomial TemplateBox[{n, m, r}, ZernikeR].

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  (25)

Numerical Evaluation  (5)

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:

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 TemplateBox[{n, m, {z, }}, ZernikeR]=TemplateBox[{n, m, z}, ZernikeR]:

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:

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_2023_zerniker, author="Wolfram Research", title="{ZernikeR}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ZernikeR.html}", note=[Accessed: 16-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_zerniker, organization={Wolfram Research}, title={ZernikeR}, year={2007}, url={https://reference.wolfram.com/language/ref/ZernikeR.html}, note=[Accessed: 16-April-2024 ]}