In[]:=
insideParams4first=FullSimplifyzExp2π
4
x
(-15
4
y
+12
2
y
2
z
-8
4
z
)+8
4
z
(-
4
y
+4
2
y
2
z
+8
4
z
)+4
2
x
(3
4
y
2
z
-4
2
y
4
z
+8
6
z
)
64
7
z
λ
/.{x->(ξ-x),y->(η-y)}
Out[]=
π8
4
z
8
4
z
+4
2
z
2
(y-η)
-
4
(y-η)
+48
6
z
-4
4
z
2
(y-η)
+3
2
z
4
(y-η)

2
(x-ξ)
+-8
4
z
+12
2
z
2
(y-η)
-15
4
(y-η)

4
(x-ξ)

32
7
z
λ

z
In[]:=
insideParams4=insideParams4first(
2
z
+
2
(-y+η)
+(-x+ξ)^2)
Out[]=
π8
4
z
8
4
z
+4
2
z
2
(y-η)
-
4
(y-η)
+48
6
z
-4
4
z
2
(y-η)
+3
2
z
4
(y-η)

2
(x-ξ)
+-8
4
z
+12
2
z
2
(y-η)
-15
4
(y-η)

4
(x-ξ)

32
7
z
λ

z
2
z
+
2
(-y+η)
+
2
(-x+ξ)
Testing for a square aperture
In[]:=
testSquare=Integrate[insideParams4,{ξ,-w/2,w/2},{η,-w/2,w/2}];//AbsoluteTiming
Out[]=
{1716.46,Null}