Difference between revisions of "ZernikeR"
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===Explicit representation=== |
===Explicit representation=== |
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| − | For integer value of parameter |
+ | For integer value of parameter \(m\) of the [[Bessel transform]], the Zernike polynomial is expressed as follows: |
| − | : |
+ | : \(\!\!\!\!\!\!\!\!\!\! (1) \displaystyle ~ ~ ~ |
R_n(x)= \sum_{k=0}{(n-m)/2} |
R_n(x)= \sum_{k=0}{(n-m)/2} |
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\frac{(-1)^k ~ (n-k)!} |
\frac{(-1)^k ~ (n-k)!} |
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{k! ~ \left( \frac{n+m}{2}-k \right) ! ~ \left( \frac{n-m}{2}-k \right) ! } |
{k! ~ \left( \frac{n+m}{2}-k \right) ! ~ \left( \frac{n-m}{2}-k \right) ! } |
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x^{n-2k} |
x^{n-2k} |
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| + | \) |
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| − | $ |
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| − | at least for integer values of |
+ | at least for integer values of \((n\!-\! m)/2\). |
<ref> |
<ref> |
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http://mathworld.wolfram.com/ZernikePolynomial.html |
http://mathworld.wolfram.com/ZernikePolynomial.html |
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</ref> |
</ref> |
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The Zernike polynomial is implemented in [[Mathematica]], and can be called as |
The Zernike polynomial is implemented in [[Mathematica]], and can be called as |
||
| − | : |
+ | : \( \rm ZernikeR[n,\nu,x]\) |
| − | where |
+ | where \(n\) is order of the polynomial, \(\nu\) is parameter of the corresponding [[Bessel transform]] and the last argument \(x\) is just argument of the Zernike polynomial. However, not all versions of Mathematica support this option. |
<!-- |
<!-- |
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| − | For fixed |
+ | For fixed \(\nu\) and \(n\), the Zernike polynomial is exposed eight elementary func |
| − | For integer values of |
+ | For integer values of \(\nu\), the [[Zernike polynomial]] is indeed polynomial. |
!--> |
!--> |
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==Examples== |
==Examples== |
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| − | For the principal mode ( |
+ | For the principal mode (\(mm=0\)), the first Zernike polynomials are |
| − | : |
+ | :\( R_0(x)=1\) |
| − | : |
+ | :\( R_2(x)=-1+2 x^2\) |
| − | : |
+ | :\( R_4(x)=1-6x^2+6x^4\) |
| − | : |
+ | :\( R_6(x)=-1+12 x^2-30 x^4+20 x^6\) |
==References== |
==References== |
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Latest revision as of 18:26, 30 July 2019
ZernikeR, or Zernike polynomial is eigendunction of the Bessel transform.
Explicit representation
For integer value of parameter \(m\) of the Bessel transform, the Zernike polynomial is expressed as follows:
- \(\!\!\!\!\!\!\!\!\!\! (1) \displaystyle ~ ~ ~ R_n(x)= \sum_{k=0}{(n-m)/2} \frac{(-1)^k ~ (n-k)!} {k! ~ \left( \frac{n+m}{2}-k \right) ! ~ \left( \frac{n-m}{2}-k \right) ! } x^{n-2k} \)
at least for integer values of \((n\!-\! m)/2\). [1] The Zernike polynomial is implemented in Mathematica, and can be called as
- \( \rm ZernikeR[n,\nu,x]\)
where \(n\) is order of the polynomial, \(\nu\) is parameter of the corresponding Bessel transform and the last argument \(x\) is just argument of the Zernike polynomial. However, not all versions of Mathematica support this option.
Examples
For the principal mode (\(mm=0\)), the first Zernike polynomials are
- \( R_0(x)=1\)
- \( R_2(x)=-1+2 x^2\)
- \( R_4(x)=1-6x^2+6x^4\)
- \( R_6(x)=-1+12 x^2-30 x^4+20 x^6\)