Zernike polynomials

In mathematics the Zernike polynomials, named after Frits Zernike, are a sequence of orthogonal polynomials which play an important role in geometrical optics.

Related Topics:
Mathematics - Frits Zernike - Sequence - Orthogonal - Polynomial - Optics - Wavefront

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There are even and odd Zernike polynomials. The odd Zernike polynomials are defined as

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:Z^{m}_n( ho,phi) = R^m_n( ho),cos(m,phi) !

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and the even Zernike polynomials as

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:Z^{-m}_n( ho,phi) = R^m_n( ho),sin(m,phi), !

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where m and n are nonnegative integers with ngeq m, phi is the azimuthal angle in radians, and ho is the normalized radial distance. The radial polynomials R^m_n are defined as

Related Topics:
Integer - Azimuth - Angle - Radian

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:R^m_n( ho) = ! sum_{k=0}^{(n-m)/2} !!! rac{(-1)^k,(n-k)!}{k!,((n+m)/2-k)!,((n-m)/2-k)!} ; ho^{n-2,k} quadmbox{if } n-m mbox{ is even}

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and R^m_n( ho)=0 if n-m is odd.

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