where and are non-negative integers, for which: . is the azimuthal angle and is the normalized radial distance.
${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle n \ geq m}$${\ displaystyle \ phi}$${\ displaystyle \ rho}$

The radial polynomials are defined according to
${\ displaystyle R_ {n} ^ {m}}$

[For purists it should be pointed out that in physics and optics these functions of two arguments are called polynomials, but depending on the application, only the radial component, i.e. the sine-cosine-shaped azimuth functions, are regarded as too trivial to allow a name extension such as effect on Zernike functions .]

Rotating the coordinate system by the angle does not change the value of the polynomial:
${\ displaystyle \ alpha = 2 \ pi / m}$

${\ displaystyle G ^ {m} (\ phi + \ alpha) = G ^ {m} (\ phi) \ !.}$

The radius-dependent part is a polynomial over of degree , which does not contain a power smaller . is an even (odd) function if is even (odd).
${\ displaystyle \ rho}$${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle R_ {n} ^ {m}}$${\ displaystyle m}$

The radius-dependent part represents a special case of the Jacobi polynomials .
${\ displaystyle P_ {n} ^ {(\ alpha, \ beta)} (z)}$

For some years now, the use of Zernike polynomials has also been common in optometry and ophthalmology . Here, deviations of the cornea or lens from the ideal shape lead to imaging errors.