Function Repository Resource:

# PseudoZernikeR

Evaluate the radial pseudo-Zernike polynomial

Contributed by: Jan Mangaldan
 ResourceFunction["PseudoZernikeR"][n,m,r] gives the radial pseudo-Zernike polynomial .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Explicit polynomials are given when possible.
ResourceFunction["PseudoZernikeR"] can be evaluated to arbitrary numerical precision.
ResourceFunction["PseudoZernikeR"] automatically threads over lists.

## Examples

### Basic Examples (3)

Evaluate numerically:

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Evaluate symbolically:

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Plot over a subset of the reals:

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### Scope (3)

Evaluate to high precision:

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

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Simple exact values are generated automatically:

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PseudoZernikeR threads elementwise over lists:

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### Properties and Relations (4)

Obtain the pseudo-Zernike polynomials from their generating function:

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Compare with the directly computed sequence:

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Verify an expression for the pseudo-Zernike polynomial in terms of the Jacobi polynomial JacobiP:

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Verify a recurrence relation for the pseudo-Zernike polynomials:

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Verify the orthogonality relation for the pseudo-Zernike polynomials:

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### Neat Examples (1)

Visualize pseudo-Zernike polynomials over the unit disk:

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## Version History

• 1.0.0 – 06 February 2023