## Abstract

In our previous paper [Appl. Opt. **51**, 8490 (2012)] we considered the Zernike polynomials for a unit annular ellipse aperture. In that paper many equations were used and were solved by MATLAB language and by hand, and many times these rewritten equations had some written mistakes. In the Diaz and Mahajan comment [Appl. Opt. **52**, 5962 (2013)] on the work, some remarks were true and others were not. In this reply, we will discuss their comment in detail.

© 2013 Optical Society of America

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### Equations (3)

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(1)
$${Z}_{2}^{\prime}(x,y)={Z}_{2}(x,y)-\frac{{\int}_{-1}^{1}{\int}_{-b\sqrt{1-{x}^{2}}}^{b\sqrt{1-{x}^{2}}}{Z}_{1}(x,y){Z}_{2}(x,y)\mathrm{d}y\mathrm{d}x-{\int}_{-k}^{k}{\int}_{-b\sqrt{k-{x}^{2}}}^{b\sqrt{k-{x}^{2}}}{Z}_{1}(x,y){Z}_{2}(x,y)\mathrm{d}y\mathrm{d}x}{{\int}_{-1}^{1}{\int}_{-b\sqrt{1-{x}^{2}}}^{b\sqrt{1-{x}^{2}}}{Z}_{1}^{2}(x,y)\mathrm{d}y\mathrm{d}x-{\int}_{-k}^{k}{\int}_{-b\sqrt{k-{x}^{2}}}^{b\sqrt{k-{x}^{2}}}{Z}_{1}^{2}(x,y)\mathrm{d}y\mathrm{d}x}*{Z}_{1}(x,y).$$
(2)
$$g=[1,x,y,2*({x}^{2}+{y}^{2})-1,{x}^{2}-{y}^{2},2*x*y,3*x*({x}^{2}+{y}^{2})-2*x,3*y*({x}^{2}+{y}^{2})-2*y,6*{({x}^{2}+{y}^{2})}^{2}-6*({x}^{2}+{y}^{2})+1,{x}^{3}-3*x*{y}^{2},3*{x}^{2}*y-{y}^{3},4*{x}^{3}*y-4*x*{y}^{3},8*x*y*({x}^{2}+{y}^{2})-6*x*y,4*{x}^{4}-3*{x}^{2}-4*{y}^{4}+3*{y}^{2},{x}^{4}-6*{x}^{2}*{y}^{2}+{y}^{4}]$$
(3)
$$C4=\sqrt{\frac{12}{({k}^{2}{b}^{2}(3{b}^{2}{k}^{2}-12{k}^{2}-8)+3{k}^{4}+3{b}^{4}+3-2{b}^{2})}.}$$